Notes on Markov embedding
Michael Baake (Bielefeld), Jeremy Sumner (UTAS, Hobart)
TL;DR
This paper revisits the representation problem of finite-dimensional Markov matrices in Markov semigroups, focusing on algebraic properties and criteria for specific subclasses like circulant and doubly stochastic matrices.
Contribution
It provides new criteria and insights into the embedding problem for various subclasses of Markov matrices, emphasizing algebraic structures and the centraliser connection.
Findings
Criteria for embedding of specific matrix subclasses
Analysis of algebraic properties related to the embedding problem
Connection between Markov matrices and their centralisers
Abstract
The representation problem of finite-dimensional Markov matrices in Markov semigroups is revisited, with emphasis on concrete criteria for matrix subclasses of theoretical or practical relevance, such as equal-input, circulant, symmetric or doubly stochastic matrices. Here, we pay special attention to various algebraic properties of the embedding problem, and discuss the connection with the centraliser of a Markov matrix.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Markov Chains and Monte Carlo Methods · Bayesian Methods and Mixture Models
Notes on Markov embedding
Michael Baake
Fakultät für Mathematik, Universität Bielefeld,
Postfach 100131, 33501 Bielefeld, Germany
and
Jeremy Sumner
School of Mathematics and Physics, University of Tasmania,
Private Bag 37, Hobart, TAS 7001, Australia
Abstract.
The representation problem of finite-dimensional Markov matrices in Markov semigroups is revisited, with emphasis on concrete criteria for matrix subclasses of theoretical or practical relevance, such as equal-input, circulant, symmetric or doubly stochastic matrices. Here, we pay special attention to various algebraic properties of the embedding problem, and discuss the connection with the centraliser of a Markov matrix.
Key words and phrases:
Markov matrix, embedding problem, semigroup, centraliser
1991 Mathematics Subject Classification:
60J10
1. Introduction
A stochastic or Markov matrix is a matrix with non-negative entries and row sums , which is the convention we use here. We call positive when all its entries are positive, and the zero matrix is singled out from the non-negative matrices by calling it trivial. A Markov generator , also known as a rate matrix, has non-negative entries off the diagonal and row sums [math], and is the homogeneous Markov semigroup (or monoid, to be more precise) generated by ; see [7] for general background. It is an old and still only partially resolved question which Markov matrices are embeddable, meaning that they appear in a Markov semigroup. For a given , this is clearly equivalent to the existence of a rate matrix such that .
While the traditional focus was on irreducible Markov matrices, recent progress on models of biological evolution has put new emphasis also on reducible and even on absorbing Markov chains, and on submodels with additional algebraic structure; see [2, 29, 15, 35] for recent investigations. Moreover, an answer to the embedding problem gives additional insight into Markov semigroups that are being used in many places; see [3] and references therein for recent examples. This motivates us to revisit the embedding problem with an eye on more specific classes of Markov matrices. Here, we only consider the finite-dimensional case, which gives nice relations to rather helpful algebraic structures. Nevertheless, it is inevitable to get in contact with the topologically complicated structure of the boundary of the set of embeddable matrices (as detailed in Kingman’s influential paper [27] on the subject), which can be seen as one of the reasons why the problem has not yet received a complete solution.
An efficient starting point to the embedding problem is the paper by Davies [10], who collected a good number of known results and examples in one place, with a useful list of references. We will refer to it frequently. Also, [22] gives an insightful overview of many of the early results. Some care is required in identifying the precise conditions in informal statements, which are often implicit, such as irreducibility or a positive determinant. One should note that many of the early results give abstract characterisations that are of limited use in practice. Our goal here is to revisit the embedding problem from a slightly different perspective, where we treat it in a more concrete fashion for various classes of matrices that are natural from an algebraic point of view or that frequently show up in applications.
The paper is organised as follows. In Section 2, we set the scene, recall some of the known results, and formulate various algebraic and asymptotic properties for later use. The classic case of two-dimensional () Markov matrices is reviewed in Section 3, where we select calculations and proofs with an eye to our later needs in higher dimensions. One generalisation, the class of equal-input matrices, is then treated in Section 4, where an interesting dichotomy between even and odd dimensions shows up. Section 5 presents results for the class of circulant matrices, which have a rich theory of their own. Finally, we discuss various more specialised classes of Markov matrices for in Section 6, once again aiming at more concrete criteria, and close with a brief outlook in Section 7.
2. General setting and results
Let us begin by recalling some necessary (but generally far from sufficient) conditions for embeddability. For convenience, we give brief hints on the proofs or references. We use to denote the spectrum of a matrix , usually including multiplicities. When the latter are not important, we simply consider as a set. Let us also recall that the -dimensional Markov matrices form a closed, convex subset , which has locally constant, topological dimension , where we generally assume to avoid trivialities.111Here and below, we use to denote the ring of -matrices over the field . Clearly, is a monoid with respect to matrix multiplication, where the extremal elements are the stochastic -matrices [24]. Also, if is Markov, one has , with equality if and only if is a permutation matrix for an even permutation. The last property follows from general Perron–Frobenius theory applied to together with the fact that Markov matrices with determinant are diagonalisable [17, Sec. 13.6].
Proposition 2.1**.**
If a Markov matrix is embeddable, so with a Markov generator, satisfies the following properties.
- (1)
The spectra are related by . 2. (2)
One has , so , and only for . 3. (3)
If with , then . 4. (4)
Each real with must have even algebraic multiplicity. 5. (5)
* is either reducible, or positive and thus also primitive.* 6. (6)
If and , then also .
Proof.
Claim (1) is clear from the spectral mapping theorem, while (2) follows from the identity with , where means .
Property (3) is Elving’s theorem [12], compare [10, Prop. 8], while Property (4) is shown in [10, Prop. 2]. The remaining claims follow from standard results on the structure of for ; see [30, Thm. 3.2.1] for a general statement. ∎
Let us note in passing that the difficulty of solving for consists in the existence of a logarithm of that has the positivity properties required for a generator, where the latter (known as the Metzler property) is the harder constraint by far when .
Example 2.2**.**
While is trivially embeddable, any irreducible Markov matrix that is embeddable must actually be primitive and positive. For instance, is irreducible, but not primitive, while with is primitive, but not positive, so neither of these matrices is embeddable; compare Example 3.9 below for more.
Also, , which has spectrum , cannot be embeddable for a\in\bigl{[}\frac{1}{2},1\bigr{]}, as this violates Proposition 2.1 (2), as well as 2.1 (4) when .
Homogeneous Markov semigroups possess a well-known asymptotic property, which we recall here for convenience and later use; compare [25, Thms. 12.25 and 12.26] for closely related results. Also, some aspects of Proposition 2.1 may become more transparent this way.
Proposition 2.3**.**
Every finite-dimensional Markov generator has the following properties.
- (1)
If , one either has or . Eigenvalues of are either real or occur in complex conjugate pairs. 2. (2)
The minimal polynomial of is of the form with , which is to say that the algebraic and the geometric multiplicity of coincide. 3. (3)
If , the limit exists and is a Markov matrix with . As such, it is diagonalisable, with .
Proof.
While claim (1) follows from Proposition 2.1 (2) via the spectral mapping theorem, we prefer to give an independent argument with some additional insight. Define the number and set , with elements for all , and for all by construction. The Gershgorin circles of are , one of which must be , where is the closed disk of radius around . Clearly, we then have , and by Gershgorin’s theorem [17, Thm. 14.6], so . Since is a real matrix, this gives the first claim.
We know that , as has zero row sums. To show claim (2), assume to the contrary that the geometric multiplicity of [math] is smaller than the algebraic one. Then, there exists a row vector222Though an equivalent argument can be given with a column instead of a row vector, this version has the slight advantage that we can directly use the row sum normalisation of . such that but . For , this implies and thus for by induction. If denotes the -norm for row vectors, the matching matrix norm is the row sum norm, with because is a Markov matrix. Consequently, we get , which is bounded. In contrast, we have , which is unbounded due to and thus gives a contradiction. Therefore, the vector cannot exist.
Claim (3) is a simple consequence of Claims (1) and (2) in conjunction with the observation that the Markov property is preserved under taking limits, because is closed in . The projector property follows from
[TABLE]
which also implies the claim on the eigenvalues. ∎
One immediate consequence is the following. If is embeddable, the Markov semigroup defines a path (in ) of embeddable matrices from (included) via to (not included). Here, since and thus , one has , and itself is not embeddable. In this way, no embeddable Markov matrix is isolated, and any two embeddable matrices of the same dimension are pathwise connected.
When is a general Markov matrix (hence not necessarily embeddable), it is also true that is an eigenvalue with equal algebraic and geometric multiplicity [17, Thm. 13.10], which can be seen as a consequence of the normal form of non-negative matrices in [17, Eq. (13.70)]. However, need not converge as , because can be irreducible without being primitive; compare [25, Thms. 8.18 and 8.22]. This cannot occur for embeddable matrices, in line with Elving’s theorem, which is Proposition 2.1 (3) above.
Corollary 2.4**.**
If is an embeddable Markov matrix, exists and is again a Markov matrix. Moreover, is a generator that satisfies . As such, it is diagonalisable, with .
More generally, with denoting the unit circle, the same conclusions hold for any Markov matrix with .
Proof.
Though the claims for embeddable matrices follow from the more general ones via Elving’s theorem, we give a simple independent argument. When , one has , and the first claim follows from Proposition 2.3 (3). The generator property of is clear, while implies the relation for as well as the property of the spectrum.
For the general claim, recall the comment made before the corollary. It implies that such an has minimal polynomial with . All roots of have modulus , so convergence follows from a standard Jordan normal form argument. ∎
Let denote the set of -dimensional Markov matrices that are embeddable, and let be the (multiplicative) semigroup generated by it. In view of the dichotomy in Proposition 2.1 (5), we also introduce .
Fact 2.5**.**
* is a monoid, while is a semigroup and a two-sided ideal in .*
Proof.
Since is embeddable, is a semigroup with unit, and the first claim is clear. For the second, observe that the product of positive Markov matrices is positive, which implies the semigroup property, and that the product of a positive Markov matrix with any Markov matrix, in either order, is again a positive Markov matrix, which implies to be a two-sided ideal in . ∎
The set is relatively closed [27, Prop. 3] within , with the same topological dimension. In particular, one has the interior-closure inclusions
[TABLE]
as follows from [27, Prop. 4]. Moreover, contains various important relatively open subsets, as detailed in [10, Lemma 3 and Thm. 7]. The proofs use some standard deformation arguments, which can easily be extended to give the following result.
Fact 2.6**.**
The set of embeddable, -dimensional Markov matrices contains the following dense and relatively open subsets, namely
- (1)
the embeddable matrices without eigenvalues in ; 2. (2)
the embeddable matrices with distinct eigenvalues; 3. (3)
the embeddable matrices with Abelian centraliser in , as characterised in more detail below in Fact 2.10.
Moreover, each of these subsets has full measure in . ∎
Let us take a different perspective on the general embedding problem, which will provide a slightly simpler setting where some concrete answers are possible. Assume that we consider a (finite or infinite) set of rate matrices which generate a closed matrix algebra over with respect to matrix addition and multiplication. For and , the series
[TABLE]
converges compactly in any matrix norm, where for all , thus as well. Together with Proposition 2.3 (1) and its proof, one has the following connection.
Fact 2.7**.**
For any , the matrix is a Markov generator that commutes with and satisfies for some . If is embeddable, with say, one has . ∎
When , it is worth mentioning that, due to the Cayley–Hamilton theorem and the structure of the exponential series, one also has a representation
[TABLE]
where each is a power series with infinite convergence radius and . When is a rate matrix, this structure suggests to consider the (non-unital) matrix algebra
[TABLE]
which will become relevant shortly. Note that is a subalgebra of the matrix algebra of all real matrices with vanishing row sums,
[TABLE]
which does not contain and is thus non-unital, too. Clearly, has dimension or smaller, where the latter case occurs if has a minimal polynomial of degree less than . Also, since both and are closed subspaces of when viewed as real vector spaces, they are complete with respect to any chosen matrix norm on , which is another way to see that , for any , lies in .
Lemma 2.8**.**
Let be a Markov generator, with defining the corresponding semigroup. Set together with and as before, where is well defined by Corollary 2.4. Then, .
Proof.
Clearly, both and are generators. Since for any , its limit as , which is , lies in as well, because the latter is closed.
Now, observe
[TABLE]
where . Since is closed, we also have , and our claim follows. ∎
Next, we state a simple special case of for later use, which harvests the fact that the only nilpotent Markov generator is
Fact 2.9**.**
Let be a Markov generator, with . Then, the following properties are equivalent.
- (1)
The minimal polynomial of is with . 2. (2)
* is diagonalisable, with eigenvalues [math] and for some .* 3. (3)
The matrix algebra is one-dimensional.
Proof.
(1) (2) is standard, while (1) (3) follows from and . Now, let us assume (3), where implies for some . Via Proposition 2.3 (1), we see that implies , which would contradict . So, we have and , which tells us that is the minimal polynomial of . As is then an eigenvalue of , another application of Proposition 2.3 (1) implies , so (1) holds, and we are done. ∎
Let us return to Fact 2.7. Viewed differently, the generator of an embedding has to satisfy a commutativity condition. Generically, this will mean that is of the same ‘type’ as , and it is thus an interesting subproblem of the general embedding question to look at embeddability under a natural constraint on , as discussed in [33, 35, 15]. This can be of practical relevance, for instance in biological applications; see [32, Ch. 7] and references therein.
Since commutativity of matrices will be important in what follows, let us recall a classic result from algebra [19, Chs. III.15 –18], which is due to Frobenius. Here, we specialise it to the case of real matrices; see also [1, Thm. 5.15 and Cor. 5.16] for another presentation. For its formulation, we employ the standard eigenspace notation for and a given matrix .
Fact 2.10**.**
For , the following properties are equivalent and generic.
- (1)
The characteristic polynomial of is also its minimal polynomial. 2. (2)
The relation holds for every . 3. (3)
* is cyclic: is a basis of for some .* 4. (4)
The matrix ring is Abelian. 5. (5)
One has , the ring of polynomials in with coefficients in . ∎
Below, we will call matrices of this kind cyclic. In particular, Fact 2.10 applies to all matrices with simple spectrum, which means distinct eigenvalues. In the case of degeneracies, the existence of additional elements in , which is known as the centraliser (or the commutant) of , can be seen from a repetition of one of its Jordan blocks. In this context, the last claim of Fact 2.7 means that with implies as well as . This has the following consequence.
Lemma 2.11**.**
If is an embeddable Markov matrix that is cyclic, one has with .
Proof.
For the matrix , we are in the situation of Fact 2.10. Then, by Fact 2.7, the rate matrix must be an element of , which means . ∎
The significance of this result emerges from the following approximation concept.
Definition 2.12**.**
Consider an embeddable matrix , and set . A property of , say Property , is called stable if it satisfies the following three conditions:
- (1)
Property has a well-defined analogue for generators, which holds for and for all generators in ; 2. (2)
Property is preserved under taking limits within ; 3. (3)
can arbitrarily well be approximated (in some matrix norm) by other embeddable Markov matrices with Property that are also cyclic (in the sense of Fact 2.10).
When only (1) and (2) are satisfied, we call the property semi-stable.
As an example, think of a symmetric Markov matrix, so , where and all element of are then symmetric as well. Clearly, the limit of a convergent sequence of symmetric matrices is again symmetric. So, being symmetric is a semi-stable property. It is not a stable one though, since symmetric matrices have real eigenvalues, and the existence of negative eigenvalues with even multiplicity can cause a problem, as we shall see. It is thus natural to also assume that has non-negative spectrum. Then, it is the limit of sufficiently many convergent sequences of symmetric matrices, in the sense that every neighbourhood of such a matrix with non-negative eigenvalues also contains further symmetric ones that are cyclic. This can be shown by a deformation argument, and still remains true within ; see Remark 4.11 below. In contrast, the property of having simple spectrum already fails to be semi-stable, as does having geometric multiplicity for all eigenvalues.
Let us briefly comment on an algebraic consequence of Definition 2.12. When Property is a linear property (such as being symmetric), semi-stability means that the -generators span a Jordan algebra. This follows from the simple observation that , and having Property then implies that has Property as well.
Proposition 2.13**.**
Let be an embeddable Markov matrix. If has a stable property, say Property , can be embedded as with a generator having Property . If, in addition, is cyclic, every generator with must be an -generator.
Proof.
If itself is cyclic, its centraliser is Abelian, and we are in the situation of Lemma 2.11, so the first claim is clear in this case. Otherwise, Condition (3) of Definition 2.12 implies that there is a sequence of embeddable, cyclic Markov matrices with Property , say , such that .
Now, for all , we have with by Lemma 2.11, where all have Property due to Condition (1) of Definition 2.12. The set of solutions of with a Markov generator is discrete and finite, see Fact 2.14 below, and the exponential map is locally a homeomorphism. Consequently, there is a subsequence of indices such that converges, where is a generator with property . By construction and Condition (2) of Definition 2.12, we get as claimed.
The second claim is now another consequence of Lemma 2.11 and Fact 2.10. ∎
Notice that, by [27, Prop. 4], the relative interior of as a subset of is non-empty, which means that the topological dimension of is the maximal one, namely . This will actually help to identify when Condition (3) of Definition 2.12 is satisfied.
Let us also note the following variant of [10, Cor. 10]. The proof is essentially the same, with the only difference coming from the more general commutativity result in Fact 2.10, that is, using cyclic rather than simple matrices, and thus extends [24, Thm. 1.1].
Fact 2.14**.**
Let be a Markov matrix that is cyclic. Then, the solutions of form a discrete set in , and these solutions commute with one another and with . Moreover, only a finite number of the solutions can be Markov generators. ∎
Finally, let us recall a diagonalisability result that will come in handy later.
Fact 2.15**.**
Let . If is diagonalisable, then so is .
Proof.
Via the Jordan–Chevalley decomposition, can be written as with diagonalisable, nilpotent, and . In particular, the minimal polynomial of is , for some . Then, where is still diagonalisable, while , with again being nilpotent.
Since is diagonalisable, and with , the matrix is diagonalisable as well, and we must have , as this is the only diagonalisable nilpotent matrix. But this implies , as any larger would give a contradiction to being the minimal polynomial of . This means together with diagonalisable. ∎
Remark 2.16**.**
In terms of any explicit Jordan form of versus , Fact 2.15 can also be seen as a consequence of [18, Thm. 1.36 (a)], since the derivative of has no zero in .
We are now set to embark on the Markov embedding problem.
3. Two-dimensional Markov matrices
The situation for is particularly simple, because both eigenvalues are real and the determinant condition from Proposition 2.1 (2) actually is sufficient to give a complete characterisation of , via . We recall this well-known result and give a short explicit proof, both for the convenience of the reader and for later use; for background on the underlying calculations around matrix functions, we refer to [18].
Theorem 3.1** (Kendall; see [27]).**
For a Markov matrix with , the following statements are equivalent.
- (1)
* is embeddable.* 2. (2)
. 3. (3)
, which means . 4. (4)
* has positive spectrum, that is, positive eigenvalues only.*
Proof.
All Markov matrices for are covered by the parametrisation chosen. The equivalence of (2), (3) and (4) is elementary, where (2) is a necessary condition for embeddability by Proposition 2.1 (2), so (1) (2) (3) (4) is clear.
Now, means . The embedding for the trivial case is , wherefore we now assume . The generator has spectrum . Consequently, is its spectral radius, and
[TABLE]
converges in norm. Since , one finds
[TABLE]
which is a Markov generator because the scalar prefactor of is always a positive number in our case. Since by construction, any such is embeddable, and we see that the necessary determinant condition is also sufficient. ∎
Remark 3.2**.**
Our proof of Theorem 3.1 was chosen for later generalisability. When and , a simple alternative argument could employ the diagonalisability of as
[TABLE]
which directly gives (5) by the definition of the logarithm of a matrix. This more directly shows that is embeddable if and only if it has positive spectrum.
It is instructive, and will be useful later on, to also consider the complementary point of view to the proof of Theorem 3.1, where one starts from a general rate matrix with and calculates . We may assume to avoid the trivial case . Now, with , one explicitly calculates the exponential series as
[TABLE]
which also covers the limit \alpha+\beta\,\mbox{\footnotesize\searrow}\,0 via l’Hôpital’s rule. Since the sum of the two off-diagonal elements of satisfies for all , the spectral radius of is less than , and one can verify that all Markov matrices from Theorem 3.1 are covered precisely once. This gives the following result that is specific to ; see [8, 9] for some results with , and [10, Cor. 10 (3) and Thm. 11] for a summary.
Corollary 3.3**.**
If a two-dimensional Markov matrix is embeddable, there is precisely one rate matrix such that , namely the one from Eq. (5). ∎
Remark 3.4**.**
Note that Theorem 3.1 is not restricted to irreducible Markov matrices. It is worth noting that the irreducible cases are also reversible, and that the embeddable ones among them are also reversibly embeddable, as discussed in more generality in [5]. Reversible Markov matrices form another interesting class, though we do not consider them here.
The property of reversibility fails to be semi-stable, as can be seen from for small , which is both embeddable and reversible, while the limit \varepsilon\hskip 0.5pt\hskip 0.5pt\raisebox{1.0pt}{\text{\tiny\searrow}}\hskip 0.5pt\hskip 0.5pt0 gives the matrix , which is neither. However, if is a converging sequence of reversible, irreducible Markov matrices such that the limit is still irreducible, reversibility is preserved as well.
Although is not really representative for the general case, it is nevertheless instructive to illustrate the situation from the viewpoint of convex sets, which we show in Figure 1. The algebraic situation for is as follows.
Lemma 3.5**.**
One has , while consists of all elements of with .
Proof.
If and are embeddable, we have by Theorem 3.1. Then, as well, which means also is embeddable, and is closed under multiplication. The second claim follows from Proposition 2.1 (5). ∎
To get a better understanding in the matrix setting, we take a look at commutativity. Writing and as before, and setting and , a simple calculation gives the following result.
Fact 3.6**.**
One has . ∎
When and are embeddable, so and , one has , where if . Otherwise, the new rate matrix can be calculated by the Baker–Campbell–Hausdorff (BCH) formula,333We refer to the WikipediA entry on the BCH formula for a quick summary. at least in principle. We shall give a simple, closed formula for this case below in Eq. (15). One consequence is that belongs to the Lie algebra generated by and , which motivated the investigation of Lie–Markov models [33, 35]. While the row sums of all vanish, it remains a difficult problem (when ) to decide when is a Metzler matrix, and hence a Markov generator.
In general, when and are embeddable but fail to commute, the embedding semigroups have no simple relation to one another. The situation, neither restricted to nor to Markov generators, reads as follows; see [13] for background and various extensions.
Lemma 3.7**.**
For , with , the following statements are equivalent.
- (1)
One has with . 2. (2)
One has for all . 3. (3)
One has for all . 4. (4)
One has for some and all . 5. (5)
One has for some , and all .
Proof.
Condition (2) follows from (1), as can be checked by a simple calculation with the exponential series, and (2) obviously implies (3), (4) and (5), where (3) (4) is also clear.
Now, assume for small . Evaluating the time derivative of both sides at gives . With this, now evaluating the second time derivative of both sides at , one sees that and must commute, so (4) (1) (2).
Finally, assume (5). As is covered by (4), we may take without loss of generality, by rescaling all three matrices if necessary. This time, evaluating the derivative at gives and thus the double identity
[TABLE]
because is invertible. The analogous exercise with the second derivative leads to
[TABLE]
where the second equality emerges from Eq. (7) via multiplying it by from the left. This implies , and Eq. (7) then gives . This, in turn, implies , hence (5) (1), and we are done. ∎
Remark 3.8**.**
Lemma 3.7 is related to rate matrix estimates in phylogenetic reconstructions. There, a Markov matrix for time needs to be consistent with added data or measurements at an intermediate time , meaning that should hold for suitable generators , and . When this is to remain true for some small intervals around and , which may be viewed as some kind of stability requirement, arguments as in the previous proof force the three generators to be equal. There are variants of this situation, the details of which are left to the interested reader. Particular aspects, such as the inconsistency of some time-reversible models and consequences thereof, are discussed in [34].
Let us next look at an interesting example where a power of a non-embeddable matrix is embeddable, and how this emerges.
Example 3.9**.**
Consider from Example 2.2, which is primitive because
[TABLE]
is positive, while itself is not. Consequently, is not embeddable, by Proposition 2.1 (5), while certainly is, by an application of Theorem 3.1.
Using the formula from Eq. (5), one finds with
[TABLE]
where is another matrix root of , but a positive and embeddable one.
The analogous phenomenon happens whenever a Markov matrix fails to be embeddable, though a power of it is. Recall that the embeddable Markov matrices are the non-singular Markov matrices that possess a Markov -th root for every ; see [27, 24].
Figure 1 also illustrates and its subset of symmetric Markov matrices, some aspects of which will later be extended in Remark 4.11. With hindsight, the set can be related to different matrix algebras, including . While this viewpoint becomes highly complex for , the situation is simpler for another matrix algebra, which we shall discuss next.
4. Equal-input matrices
Following [32, Sec. 7.3.1], let us consider a special, but practically important, class of Markov matrices, known as equal-input (or Felsenstein [14]) matrices. For its formulation, let be a -matrix with equal rows, each being , and define as its summatory parameter, which will also be called its parameter sum. A matrix with and all is Markov, but of rank , and thus never embeddable for .
Fact 4.1**.**
Any square matrix with equal rows and summatory parameter satisfies the relation . When , it is always diagonalisable, while it is nilpotent for . In the latter case, it is diagonalisable if and only if . ∎
Now, since itself is not interesting enough, consider
[TABLE]
which is a matrix with row sums . It is Markov if and for all . We call such Markov matrices equal-input, since they describe a Markov chain where the probability of a transition , for , depends only on . As such, an equal-input Markov matrix emerges from a matrix with equal rows as given in Eq. (8). We denote the set of all equal-input Markov matrices with fixed dimension by , so .
Since has spectrum , with copies of [math], one gets
[TABLE]
which, for embeddability, has to lie in by Proposition 2.1 (2). When is even, Eq. (9) then clearly implies , where with all means and thus . Note that coincides with the case treated in Theorem 3.1, so equal-input Markov matrices can be viewed as one generalisation of Section 3 to higher dimensions.
For general , whenever , the spectral radius of is less than and we can use the relation to obtain
[TABLE]
in complete analogy to Eq. (5), where is a generator of equal-input type, called an equal-input generator from now on, as it is derived from a matrix with equal rows in the obvious way. Note that means , so that the formula in Eq. (10) also covers this limiting case via l’Hôpital’s rule.
Observe that being of equal-input type is a semi-stable, but not a stable property in the sense of Definition 2.12, so we should expect some subtleties. Thus, consider a general matrix with equal rows . Then, with is an equal-input generator whenever all . With , one gets
[TABLE]
this time in complete analogy to the formula in Eq. (6). In particular, all equal-input Markov matrices with are embeddable in this way.
Also, observing that
[TABLE]
in obvious notation, we know that the product of equal-input Markov matrices is again equal-input [32, Lemma 7.2 (iv)]. Also, implies , in line with the fact that the equal-input Markov matrices form the (closed) positive cone of a matrix algebra.
Putting these pieces together, we have the following result for even dimensions.
Proposition 4.2**.**
For even, an equal-input Markov matrix is embeddable if and only if its parameter sum satisfies . The set of such matrices, , forms a monoid. The irreducible elements in are the positive ones, and they form a semigroup and a two-sided ideal within . ∎
For odd and , the eigenvalues of are and , where the algebraic multiplicity of is and thus even. Hence, the case can neither be excluded from embeddability by the determinant criterion nor by Proposition 2.1 (4); compare [10, Prop. 2]. To illustrate this point, let us review and expand [10, Ex. 16] in our setting.
Example 4.3**.**
Consider the commuting Markov generators
[TABLE]
with spectra \sigma(Q)=\big{\{}0,-\frac{1}{2}\bigl{(}3\pm\mathrm{i}\hskip 0.5pt\sqrt{3}\,\bigr{)}\big{\}} and . With , one gets
[TABLE]
The matrix ring is two-dimensional (viewed as a vector space over ). Now, fails to have simple spectrum, but is diagonalisable, so is larger than by Fact 2.10. If is a rate matrix, a simple computation shows that forces to be doubly stochastic, which means that also all column sums of are zero.
Via an explicit calculation, one finds the embeddable matrix
[TABLE]
with and . The particular property here is that is a symmetric, equal-input Markov matrix which has a negative eigenvalue with algebraic multiplicity , , but summatory parameter . This shows that Proposition 4.2 does not extend to odd. Note also that with a symmetric is impossible because would give a contradiction with .
Algebraically, we have , where is doubly stochastic and circulant, see Fact 4.4 below for more, but neither symmetric nor of equal-input type. Since , a one-parameter family of such examples can be obtained by using the generator with sufficiently small .
Now, consider a small neighbourhood of in . Since , where is a set of full topological dimension , the local homeomorphism property of the exponential map implies that must contain other embeddable equal-input matrices with , where not all are equal. None of them can be equal-input embeddable. In this sense, the above example is neither isolated nor restricted to a lower-dimensional family of matrices.
The observations on from Example 4.3 can be summarised and extended as follows. A generalisation to arbitrary will be stated in Lemma 4.10.
Fact 4.4**.**
A matrix commutes with the Markov generator from Eq. (12) if and only if there is a real number such that all row and all column sums of equal . The centraliser of is five-dimensional, and any is of the form
[TABLE]
with . In particular, a Markov matrix or a Markov generator commutes with if and only if it is doubly stochastic.
Proof.
The first claim follows from a simple calculation around , which also gives the parametrisation chosen. The last claim is obvious via restricting to the set of Markov matrices or generators, respectively. ∎
Clearly, the matrix from Example 4.3 cannot be written as with an equal-input generator, as any then has parameter sum , while we had . Also, due to Eqs. (10) and (11), equal-input embeddable Markov matrices are closed under multiplication, and we have the following counterpart of Proposition 4.2.
Lemma 4.5**.**
When is odd, an equal-input Markov matrix is equal-input embeddable if and only if , and the class of such matrices forms a monoid. ∎
The difference to even is that there are other embeddable cases, but not with a generator of equal-input type. This matches with the fact that being equal-input is only a semi-stable property in the sense of Definition 2.12. We thus have the following general statement.
Theorem 4.6**.**
An equal-input Markov matrix is equal-input embeddable if and only if its parameter sum satisfies . The class of such matrices forms a monoid, with the subset of positive ones being a two-sided ideal in this monoid.
Moreover, when is even, irrespectively of the type of generator, no further equal-input Markov matrices are embeddable, while additional embeddable cases do exist for odd, where then is a negative eigenvalue with even algebraic multiplicity. ∎
When is equal-input embeddable, this does not exclude further embeddings of a different nature, as we shall see in Remark 6.7. However, more can be said about the equal-input solutions. If is an equal-input generator, one has and thus
[TABLE]
which means that the summatory parameter of is . Now, let and be equal-input generators, with parameters and . When , it is immediate that , and Eq. (14) then implies . This gives us the following uniqueness result, which is a generalisation of Corollary 3.3.
Corollary 4.7**.**
If an equal-input Markov matrix is embeddable as with a generator of equal-input type, which is possible if and only if , the generator is unique. For , it is given by Eq. (10), and by for . ∎
One can further exploit the monoid property of embeddable equal-input Markov matrices. If and are equal-input generators, one finds with the new generator
[TABLE]
where the second line follows from the identity . When , which is equivalent to , Eq. (15) simplifies to , in line with Lemma 3.7. In any case, the summatory parameters of the exponentials are related by
[TABLE]
in accordance with Eq. (11).
Remark 4.8**.**
The class of equal-input Markov matrices contains a subclass, defined by the condition , called constant-input Markov matrices. For , they in particular cover the Jukes–Cantor mutation matrices [21]. Constant-input Markov matrices are closed under multiplication, so the previous analysis can be restricted to this subclass. Since from Example 4.3 is a constant-input Markov matrix, we get the result of Theorem 4.6 also for this subclass, with the same distinction between even and odd. We shall meet this class again in Section 6.2.
Let be as above, with summatory parameter together with for all , and consider the equal-input generator . Since when , let us assume . Then, we have and , so has minimal polynomial . In particular, is diagonalisable, with eigenvalues [math] and , the latter with multiplicity . In this context, one has the following extension of Fact 2.9.
Corollary 4.9**.**
Let be a Markov generator that satisfies any of the equivalent properties of Fact 2.9, say with . Then, [math] is a simple eigenvalue of if and only if is an equal-input generator with summatory parameter .
Proof.
Under the assumptions, we have . When is simple, with eigenvector , we see that each column of must be a scalar multiple of . If the -th column is with , we have , which is a generator if and only if all together with . Since [math] is simple, we must have .
The converse direction was shown above. ∎
For , an equal-input generator inevitably has multiple eigenvalues such that its minimal and characteristic polynomials differ. Consequently, by Fact 2.10, the centraliser is non-Abelian. Interestingly, on the level of generators, one has the following elementary result, which can be seen as a generalisation of Fact 4.4.
Lemma 4.10**.**
Let be an equal-input generator, with non-negative parameters and . Then, a Markov generator commutes with if and only if . When , this is equivalent to saying that is an equilibrium state of .
Proof.
When , which means and , every generator commutes with and the statement is trivial. Next, assume and . A matrix commutes with if and only if it commutes with . When is a generator, one has because has constant columns. Then, means , each row of which is the condition stated. Since is a probability vector for , the last claim is clear. ∎
Notice that such a generator may have several equilibrium states, which form a convex set. Then, when , the characterisation from Lemma 4.10 specifies only one of them.
Remark 4.11**.**
There is a doubly stochastic, constant-input Markov matrix, with parameters , which has several interesting properties. Clearly, , so is not embeddable, but lies on the boundary of . This is also clear from the relation with being any irreducible, doubly stochastic generator in dimensions. More importantly, as can be shown by the methods used for [23, Thm. 2.7], the subsets of of equal-input or of certain doubly stochastic matrices are star-shaped with respect to , which is visible in Figure 1 for . Since is also circulant and symmetric, this will be useful for later deformation arguments.
5. Circulant matrices
A -matrix is called circulant if each of its rows emerges from the previous one by cyclically shifting it one position to the right (see below for more). Such matrices have a rich theory of their own; see [11] for a detailed exposition. Here, we are interested in circulant matrices that are also Markov matrices or generators, respectively. The interested reader may consult [32, Ch. 7.3.2] for how these matrices fit into the setting of ‘group-based models’, and [36] for an extension to ‘semigroup-based models’. The latter classification in particular includes the equal-input models discussed in Section 4.
For , a Markov matrix is circulant precisely when it is a Markov matrix of constant-input type, which means . By Theorem 3.1, such an is embeddable if and only if . In fact, with , one can rewrite our earlier formula as
[TABLE]
In particular, whenever a circulant with is embeddable, Corollary 3.3 together with Eq. (5) implies that this is only possible with a circulant generator, which also fits Proposition 2.13. The explicit computations rely on the formula , which can be seen as a decomposition of the exponential function over the cyclic group .
Let us look at circulant matrices in more generality. By [11, Thm. 3.1.1], is circulant if and only if commutes with the permutation matrix
[TABLE]
which is the standard -dimensional representation of the cyclic permutation . Since has simple spectrum, namely , and since is also a doubly stochastic matrix, as is any power of it, one finds the following consequence.
Fact 5.1**.**
The circulant matrices in , respectively in , are the elements of the Abelian matrix ring , respectively . The -dimensional, circulant Markov matrices are precisely the convex combinations of , which form a simplex of dimension within , and also a monoid under matrix multiplication. In particular, every circulant Markov matrix is doubly stochastic. ∎
Clearly, the exponential of a circulant matrix is again circulant. Moreover, being circulant is a stable property. This can be seen by a deformation argument on the basis of Remark 4.11 as follows. Given , select a circulant matrix with simple spectrum near the matrix from Remark 4.11 such that is embeddable for all , which is possible, and apply [10, Cor. 6] to get the approximation property. By Proposition 2.13, we thus get the following consequence.
Corollary 5.2**.**
Any embeddable circulant Markov matrix is circulant-embeddable. When its centraliser is Abelian, it is only circulant-embeddable. ∎
For , let be the cyclic group of order , represented as with addition modulo , and let with and define the characters of , which satisfy the well-known orthogonality relations [20, Thm. 16.4]
[TABLE]
With this, some elementary calculations establish the following result.
Fact 5.3**.**
Let . Then, the functions with , defined by
[TABLE]
satisfy the following properties.
- (1)
For any , one has the decomposition . In particular, this gives for . 2. (2)
For any , the function possesses a globally convergent Taylor series, namely , and is thus real-valued. ∎
Due to the first property, this can be considered as a decomposition of the exponential function over the cyclic group , thus generalising the earlier case, . If , so with , one also has the summatory identity
[TABLE]
for . We shall return to these functions after an explicit treatment of and .
5.1. A cyclic model for
Let be the matrix from Eq. (17) for ,
[TABLE]
which has spectrum with a primitive third root of unity, so and . Also, by Fact 2.10, one has . Now, is diagonalisable as with the unitary Fourier matrix444Note that our actually is the complex conjugate of the matrix used for discrete Fourier transform.
[TABLE]
Next, define two Markov generators as , which equals from Eq. (12), and . They satisfy
[TABLE]
and thus generate a two-dimensional matrix algebra over , which is an Abelian subalgebra of from Eq. (4). This subalgebra consists of the matrices
[TABLE]
with , which are all circulant. Since the rate matrices and commute, one has , wherefore it is natural to set
[TABLE]
Now, with and , we get
[TABLE]
with x(\alpha,\beta)=\frac{1}{3}\bigl{(}1+2\hskip 0.5pt\hskip 0.5pt\mathrm{e}^{-\gamma}\cos\bigl{(}\delta-\frac{2\pi}{3}\bigr{)}\bigr{)} and , where and . Note that x(\alpha,\beta)+y(\alpha,\beta)=\frac{2}{3}\bigl{(}1-\hskip 0.5pt\mathrm{e}^{-\gamma}\cos(\delta)\bigr{)}. Another way to write the result, with , is
[TABLE]
where holds for all .
To see which Markov matrices of the form are realised by Eq. (19), we need to determine the image of under the mapping
[TABLE]
The necessary determinant condition from Proposition 2.1 (2) implies the inequality
[TABLE]
which is satisfied, but not sufficient in this case. Figure 2 illustrates the region that is covered, where the special point belongs to I_{3}=M\bigl{(}\frac{1}{3},\frac{1}{3}\bigr{)}. The region is bounded by the two curves
[TABLE]
which both start at and approach the limit point \bigl{(}\frac{1}{3},\frac{1}{3}\bigr{)} without reaching it. The entire region is filled with straight lines from the boundary points towards this special point, as parametrised by with (lower half) and with (upper half). The special role of the limit point also emerges from
[TABLE]
which shows that the limit point is the (somewhat degenerate) envelope [16] of our family of curves, where the determinant criterion is due to Leibniz and Taylor. We can summarise our derivation as follows.
Theorem 5.4**.**
For a general circulant Markov matrix , the following statements are equivalent.
- (1)
* is embeddable.* 2. (2)
* is circulant-embeddable.* 3. (3)
The parameter pair lies in the closed region defined by the bounding curves from Eq. (20), excluding the point \bigl{(}\frac{1}{3},\frac{1}{3}\bigr{)}, which parametrises the non-embeddable matrix from Remark 4.11.
Proof.
In view of our above calculations, is clear, while follows from Corollary 5.2. ∎
Remark 5.5**.**
Circulant Markov matrices with have previously been considered in some detail in [28, Sec. III]. This class is particularly interesting from the point of view of multiple embeddings of a given Markov matrix. In smaller and smaller regions of the parameter space near \bigl{(}\frac{1}{3},\frac{1}{3}\bigr{)}, one can find matrices with an increasing number of distinct embeddings. We shall return to this point in Section 6.2.
5.2. The cases
For , the most general circulant generator reads
[TABLE]
Its exponential is , where with the permutation matrix of in analogy to the previous section, together with555The validity of Eq. (21) can easily be verified by means of a standard computer algebra program, the details of which are left to the interested reader.
[TABLE]
In the limit , one has and . Taking , the parametrisation reduces to
[TABLE]
and thus covers the line from \bigl{(}0,\frac{1}{2},0\bigr{)} to \bigl{(}\frac{1}{4},\frac{1}{4},\frac{1}{4}\bigr{)}, with M\bigl{(}\frac{1}{4},\frac{1}{4},\frac{1}{4}\bigr{)}=I_{4}. Note that no point on this line is ever reached with finite parameter values. Otherwise, letting or each means (x,y,z)\xrightarrow{\quad}\bigl{(}\frac{1}{4},\frac{1}{4},\frac{1}{4}\bigr{)}, again without ever reaching this point. In fact, since
[TABLE]
we see that the above line is the (again degenerate) envelope of our family of curves in .
In general, one always has x+z=\frac{1}{2}\bigl{(}1-\hskip 0.5pt\mathrm{e}^{-2(\alpha+\gamma)}\bigr{)}. Setting , one obtains the line \big{\{}\bigl{(}0,\frac{1}{2}(1-\hskip 0.5pt\mathrm{e}^{-2\beta}),0\bigr{)}:\beta\geqslant 0\big{\}}, without reaching . One can now check that the possible values of fill a region that is bounded by three finite surface sheets of the form with
[TABLE]
where the points on the ‘seam line’ from \bigl{(}0,\frac{1}{2},0\bigr{)} to \bigl{(}\frac{1}{4},\frac{1}{4},\frac{1}{4}\bigr{)} are never reached.
We can sum up this analysis as follows, with an accompanying illustration in Figure 3.
Theorem 5.6**.**
For a general circulant Markov matrix in , the following statements are equivalent.
- (1)
* is embeddable.* 2. (2)
* is circulant-embeddable.* 3. (3)
The parameters lie in the closed region defined by the three surface sheets with the parameter regions from Eq. (22), without the points on the envelope. ∎
In general, consider and let be the order of the cyclic subgroup that is multiplicatively generated by , where and . Now, it is not hard to compute
[TABLE]
where the second representation is a consequence of Eq. (18). Define for , where . All are Markov generators, with relations . Consequently, the algebra generated by the is Abelian and of dimension .
Now, with \alpha_{0}\mathrel{\mathop{:}}=-\bigl{(}\alpha_{1}+\cdots+\alpha_{d-1}\bigr{)}, one obtains
[TABLE]
with the coefficients
[TABLE]
With Fact 5.3 (1), one finds the relation
[TABLE]
which leads to the alternative expression
[TABLE]
This can be seen as the natural generalisation of our previous calculations, in particular Eq. (16) and the formulas for and . From here, one obtains general criteria for circulant Markov matrices and their embeddability as follows.
Theorem 5.7**.**
For fixed , the most general circulant Markov matrix is of the form , with all and . It is embeddable if and only if there are non-negative numbers such that holds for all , with the coefficients from Eq. (24). ∎
One can analyse the situation further from here in various ways. For instance, one finds \det\bigl{(}\frac{\partial x_{i}}{\partial\alpha_{j}}\bigr{)}=\hskip 0.5pt\mathrm{e}^{-d\hskip 0.5pt(\alpha_{1}+\cdots+\alpha_{d-1})}, and one can determine bounding surface sheets for the parameter region as before. We leave details to the interested reader.
Remark 5.8**.**
The above analysis can also be carried out for general (Abelian) group -based models in the sense of [32, Sec. 7.3.2]. In particular, the embedding problem for the case, otherwise known as the Kimura 3 ST model [26], has recently been treated in [31]. The authors demonstrate in Example 3.1 of their work that there exist Kimura 3 ST Markov matrices that are not embeddable via a generator of this type. However, their example is embeddable with a circulant generator . Hence, this situation is in perfect analogy with our Example 4.3 above, and the explanation of the phenomenon is the same.
6. Other classes for
While we know from Proposition 2.1 that any irreducible must be positive, this is no longer the case within , as can be seen from
[TABLE]
When , the two block matrices on the left are reducible, but embeddable, while the matrix on the right is primitive, because its square is positive. However, the matrix itself still contains a zero element, so cannot be embeddable due to Proposition 2.1 (5). In particular, even if the right-hand side can be written as , with determined via the BCH formula say, has vanishing row sums, but at least one off-diagonal element will be negative. Consequently, , and the same phenomenon occurs for all . Using other canonical ways to embed into , one sees that contains non-embeddable primitive matrices with a single [math] in one off-diagonal entry, for any of the six possible choices.
Moreover, as can be seen from [10, Ex. 14], there are positive Markov matrices that satisfy the determinant condition, but are not embeddable, which shows the higher complexity for . Let us thus look at some subclasses that are related to various matrix subalgebras of . Our results can alternatively be derived via a normal form for Markov matrices, which also underlies some of the analysis in [24, 4, 6]. A spectral characterisation of embeddable matrices with simple spectrum is given in [24, Cor. 1.3], while those with spectrum and are covered by [24, Prop. 1.4]. The case with was later solved in [4, 6]. Our approach is based on the algebraic structure of (semi-)stable properties, which gives alternative insight.
Before we embark on the special cases, let us take a closer look at the underlying algebraic structure for . When the minimal polynomial of a Markov matrix has degree , we know that is cyclic, and we can investigate under the condition that the generator satisfies with , by Lemma 2.11. When the degree is , the polynomial must be , which is the trivial case . So, we only have to worry about the case where the minimal polynomial of is for some .
Lemma 6.1**.**
Let be an embeddable Markov matrix for with minimal polynomial of degree . Then, is diagonalisable, with being one-dimensional, where and . Moreover, any generator with is diagonalisable, and one of the following cases applies:
- (1)
, so for some , and . If is a simple eigenvalue of , the generator must be of equal-input type. 2. (2)
* and is a simple eigenvalue of , which implies that is simple, with , where for some with .*
In particular, is of equal-input type whenever is a simple eigenvalue of .
Proof.
Assume as stated, with eigenvalues and . This means that has minimal polynomial with , hence eigenvalues [math] and , one of them with multiplicity . In either case, both and are diagonalisable.
Clearly, satisfies , and by Fact 2.9. Now, assume that , with a generator, and define as before, where from Corollary 2.4. By Lemma 2.8, we have , which implies . In particular, is one-dimensional, while diagonalisability of follows from that of in conjunction with Fact 2.15.
As , we have . But diagonalisable means it is either simple or satisfies for some . In the latter case, since , the generators and both are non-trivial multiples of , which gives the equalities of the one-dimensional algebras. If is simple, the equal-input property of , which has [math] as a simple eigenvalue, follows from Corollary 4.9. The equal-input property is then inherited by .
When is simple, and are linearly independent, so . Then, we have with . In this situation, we must have , which is only consistent if , as by Elving’s theorem, and . Since is simple, this gives with .
As must be simple in the last case, is an equal-input generator by Corollary 4.9 again, which also completes the argument for the final claim. ∎
Let us now turn our attention to some special matrix classes for .
6.1. Symmetric matrices for
The approximation approach fails for symmetric Markov matrices with negative eigenvalue as all nearby embeddable, symmetric matrices have some negative eigenvalue with even algebraic multiplicity by Proposition 2.1 (4). As these matrices are also diagonalisable, they can never be cyclic. However, as mentioned after Definition 2.12, being symmetric with non-negative spectrum has the required approximation property, whence an embeddable, symmetric Markov matrix with can always be written as with symmetric, by Proposition 2.13. So, let us look at the general symmetric (and then automatically doubly stochastic) Markov generator
[TABLE]
with . One has with
[TABLE]
where by Cauchy–Schwarz and due to , hence , and . In particular, any symmetric generator is negative semi-definite. Moreover, is only possible for , which means and results in , which is to be compared with Eq. (13).
The rather simple structure of the eigenvalues follows easily from the observation that is an anti-circulant matrix; see [11, p. 156] for more on this matrix class. Also, eigenvectors can be computed in closed form and read
[TABLE]
which are mutually orthogonal. In Dirac notation, let and be the normalised eigenvectors derived from this, which give the three projectors
[TABLE]
with as in Remark 4.11, as can be checked by an explicit computation.
Lemma 6.2**.**
If is the generator from Eq. (25), its exponential is given by
[TABLE]
which correctly covers the limiting case s\,\text{\footnotesize\searrow}\,0, where and \hskip 0.5pt\hskip 0.5pt\mathrm{e}^{Q}=\mymathbb{1}+\bigl{(}1-\hskip 0.5pt\mathrm{e}^{-\varDelta}\bigr{)}\hskip 0.5ptJ_{3}.
Proof.
Given , which is diagonalisable, the spectrum of is \bigl{\{}1,\hskip 0.5pt\mathrm{e}^{-\varDelta+s},\hskip 0.5pt\mathrm{e}^{-\varDelta-s}\bigr{\}}. Employing the projectors from Eq. (27), one then obtains
[TABLE]
which leads to the formula by a simple calculation. The claim on the limit follows from and the fact that means and . ∎
Remark 6.3**.**
The symmetric matrices do not form a matrix algebra, because and contains symmetric matrices that do not commute. Likewise, symmetric generators do not form a Lie algebra, because [A,B]^{T}=-\bigl{[}A^{T},B^{T}\hskip 0.5pt\bigr{]}. Consequently, the corresponding symmetric Markov matrices of the form are not closed under matrix multiplication, as discusses in [33]. However, symmetric matrices do form a Jordan algebra, with . This observation turned out to be a helpful property for identifying (semi-)stable matrix classes.
Let us now consider a general symmetric Markov matrix, written as
[TABLE]
with as well as . Note that any such matrix is also doubly stochastic, and that is again anti-circulant.
Clearly, we can have , which is . More generally, let us assume that is embeddable. If , Proposition 2.1 (5) implies that , and likewise for or . So, we have either or . If , we must have and hence also , which then fails to be embeddable by Theorem 3.1, and analogously for or . In fact, whenever two of the numbers are zero, Theorem 3.1 implies that the third lies in the interval \bigl{[}0,\frac{1}{2}\bigr{)}, which gives us the following result, where the second claim can be seen constructively from another application of Eq. (5).
Fact 6.4**.**
When a symmetric Markov matrix of the form (28) fails to be positive, it is embeddable if and only if together with . ∎
The general case can be stated as follows.
Theorem 6.5**.**
Let be a symmetric Markov matrix as given in (28), with parameters and . Then, the following statements are equivalent.
- (1)
* is embeddable and has positive spectrum.* 2. (2)
* is embeddable with a symmetric generator.* 3. (3)
There are non-negative numbers such that
[TABLE]
with and as in Eq. (26).
Futhermore, there are embeddable, symmetric Markov matrices with negative eigenvalues, but none of them can be written as with symmetric.
Proof.
(1) (2) follows from the argument used in the proof of Proposition 2.13, because an embeddable, symmetric Markov matrix with positive spectrum can be approximated by cyclic, embeddable ones, while (2) (1) is clear. Now, Lemma 6.2 gives the general form of with a symmetric generator, and (3) is the resulting condition on the parameters.
The final claim follows from the matrix of Example 4.3 together with the fact that for any symmetric generator . ∎
Let us comment a little on the situation at hand. If from Eq. (28) is embeddable with a symmetric rate matrix, so with as in Eq. (25), one has
[TABLE]
with and as in Eq. (26). We can now compare the coefficients of the characteristic polynomial of with the corresponding expressions from Vieta’s formula. This gives three necessary conditions for embeddability as follows. First, in line with Proposition 2.1 (2), one has , which means
[TABLE]
Next, one finds , which is equivalent with
[TABLE]
while the third condition on the eigenvalues can be stated as
[TABLE]
All three conditions are sharp in the sense that each possible value can be realised. Note that these conditions, even when taken together, are not sufficient for the embeddability of .
Two other properties follow from Eq. (29), namely that is positive definite and that has spectral radius less than . The former, via Sylvester’s criterion, means together with , which also follows from the trace condition, and
[TABLE]
which, in this case, is not an independent condition either. With , the other property implies that the symmetric matrix
[TABLE]
is well defined, has vanishing row sums because is a generator, and satisfies . The difficult step, when starting from this formula, consists in formulating a condition on that ensures the off-diagonal elements of to be non-negative, that is, its Metzler property. No criterion in polynomial form, say, exists for this.
As we saw in Example 4.3, there are symmetric Markov matrices with negative eigenvalues that are still embeddable, but never with a symmetric generator. These cases are naturally included in the class of doubly stochastic matrices, which we consider below in Section 6.3.
6.2. Constant input matrices
Let us go back to the constant-input Markov matrices from Section 4 and Remark 4.8. Since is odd, Theorem 4.6 and its constructive proof imply that a constant-input Markov matrix is embeddable with a constant-input generator if and only if the summatory parameter of satisfies , where means . For , we have further embeddable cases, but then necessarily with doubly stochastic generators that are not of equal-input type; compare Example 4.3 and Fact 4.4.
So, assume is constant-input with and with being doubly stochastic, the most general form of which is
[TABLE]
with and . One finds with
[TABLE]
A matching set of eigenvectors can be given as
[TABLE]
where is perpendicular to .
Since we now have with and from (31), we see that must be negative, which implies that for some and results in . Since , we have
[TABLE]
for any chosen , which implies . To make sure that is a generator, we also need , hence . Together, for any chosen , this means , which takes the form for or . This upper bound is the value we saw in Example 4.3.
Corollary 6.6**.**
Let be a constant-input Markov matrix for with corresponding summatory parameter . Then, is embeddable if and only if , with a doubly stochastic generator. This is possible if and only if .
Proof.
We can proceed constructively with the choice . Choose with the generator from Example 4.3, which is the unique constant-input generator with , and the matrix
[TABLE]
which commutes with and satisfies , by a calculation similar to the one used in Example 4.3. Note that is a generator precisely when . One finds
[TABLE]
which is a constant-input Markov matrix with . With the admissible choices for , one exhausts the claimed range of the parameter . ∎
Note that some of the arguments in the last proof are related to the claims from Lemma 6.1, because in Corollary 6.6 is a constant-input generator.
Remark 6.7**.**
We have used the choice to get the maximal range. Alternatively, when using , so and , one obtains the range with another embedding solution. So, for smaller and smaller regions, one gets an increasing number of solutions to the embedding problem; see [28] for a related discussion.
In fact, due to , one has \hskip 0.5pt\mathrm{e}^{nT}=\mymathbb{1}+\bigl{(}(-1)^{n+1}+1\bigr{)}J, and this gives extra embedding solutions also for . Indeed, when is even, is a generator when , and
[TABLE]
then is a constant-input Markov matrix with parameter . Note that this produces an analogous multi-embedding phenomenon as in the previous case, but this time for the range . These extra solutions (with ) are doubly stochastic generators that are not of equal-input type.
6.3. Doubly stochastic matrices
A natural extension of symmetric Markov matrices is provided by the family of doubly stochastic ones. The latter form a closed convex set that is again a monoid. The extremal elements are the permutation matrices, compare [24], which are linearly dependent for .
There are extremal matrices for , but the corresponding monoid is four-dimensional. Indeed, a simple calculation, compare Fact 4.4, shows that any doubly stochastic matrix can be parametrised as
[TABLE]
with and , subject to the obvious constraints to make a Markov matrix, namely and . The locally constant, topological dimension clearly is , and there is only one additional parameter in comparison to Eq. (28), namely (for excess), with giving the symmetric matrices. Note that is normal if and only if . Here, has spectrum with and .
The matrix from Example 4.3 is doubly stochastic and shows that some subtle phenomena can occur. However, if a doubly stochastic Markov matrix is embeddable, so for some generator , the row vector is a left eigenvector of both and , with eigenvalue [math] for by the spectral mapping theorem because is a rate matrix; compare Proposition 2.3 (1). This implies that the generator is doubly stochastic.
Corollary 6.8**.**
A doubly stochastic Markov matrix is embeddable if and only if with a doubly stochastic generator. ∎
Note that the generator of Eq. (30) is of equal-input type if and only if together with , which means that it is then a constant-input generator.
Theorem 6.9**.**
Let be the general doubly stochastic Markov matrix in , as given in Eq. (33), with parameters and subject to and . Let denote the minimal polynomial of . Then, is embeddable if and only if one of the following situations applies.
- (1)
, which means , and thus ; 2. (2)
, which implies for some , so is diagonalisable; if has multiplicity , we have , is symmetric, and together with ; if is simple, must be a constant-input generator with , parameter sum , and with ; 3. (3)
, and there are non-negative numbers , and some , such that
[TABLE]
with and as in Eq. (31), and .
Proof.
Case (1) is trivial, while the case distinction in (2) follows from a simple calculation with the eigenvalues of and its consequences for and . When , we are back to Fact 6.4, while implies to be equal-input and doubly stochastic, hence constant-input, and the condition stated here follows from Theorem 4.6 and Corollary 6.6.
For Case (3), is either simple or has a real eigenvalue , with and algebraic multiplicity , but geometric multiplicity , and hence a non-trivial Jordan block in the Jordan normal form. When is simple, hence cyclic, we are in Case (2) of Lemma 6.1, so is diagonalisable as well and with as before. Here, and are simultaneously diagonalisable, with a matrix that derives from the eigenvectors of as given in (32). Now, we can use and compare eigenvalues, which results in the equation as stated. The further constraints guarantee the generator property of .
Finally, the remaining Jordan case is obtained as a limit of such simple matrices, which still gives the same equation for the parameters. ∎
7. Outlook
There are many aspects of the embedding problem that we have not treated or addressed here, though some were briefly mentioned in our remarks. Among them are more general results on uniqueness or multiple solutions, which becomes increasingly difficult with growing dimension, or the classification of matrix classes that are connected with Jordan or Lie algebras, because their number also increases quickly. A more complete picture should still be achievable up to , while further constraints would be needed beyond.
From the viewpoint of biological application, it seems desirable to concretely consider matrix classes for that cover the standard mutation schemes of molecular evolution, which are commonly used in bioinformatics and in population genetics. Since the number of relevant matrix classes is much larger than for , this needs a separate treatment. In this context, it would also be relevant to know the relation to inhomogeneous Markov chains, as this can cover time-dependent processes more realistically.
Another direction is the extension of the analysis to countable state Markov chains, which would require new methods and tools from functional analysis, or to sub-stochastic matrices and their generators, which show up increasingly in theoretic and applied probability. Here, the non-negativity conditions remain the same, but the row sums for sub-stochastic matrices or generators are either or , respectively. One would expect inequalities that parallel our above results, but little has been done in this direction so far.
Acknowledgements
It is our pleasure to thank F. Alberti, B. Gardner, P.D. Jarvis, H. Kösters, A. Radl and M. Steel for discussions. We also thank the organisers and participants of MAM 10 in Hobart, Tasmania, for useful hints on the problem, and two referees for their thoughtful comments that helped to improve the presentation. This work was supported by the German Research Foundation (DFG), within the SPP 1590, and by the Australian Research Council (ARC), via Discovery Project DP 180 102 215.
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