Realization of tensor-product and of tensor-Factorization of rational functions
Daniel Alpay, Izchak Lewkowicz

TL;DR
This paper explores the realization of tensor-products and tensor-factorizations of rational functions, providing explicit formulas and methods to handle their state space representations, especially under specific limiting conditions.
Contribution
It introduces explicit formulas for tensor-factorization of rational functions and extends classical realization methods to tensor-product cases with dimension inflation.
Findings
Explicit realization formulas for tensor-products of rational functions.
A method for tensor-factorization under certain limit conditions.
Extension of classical realization theory to tensor-product scenarios.
Abstract
We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Then, under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, an explicit formula for a tensor-factorization of this function, is introduced.
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Taxonomy
TopicsMatrix Theory and Algorithms · Tensor decomposition and applications · Numerical Methods and Algorithms
realization of tensor-product and of
Tensor-Factorization of rational functions
Daniel Alpay
(DA) Faculty of Mathematics, Physics, and Computation
Schmidt College of Science and Technology
Chapman University
One University Drive Orange, California 92866
USA
and
Izchak Lewkowicz
(IL) Department of electrical engineering Ben-Gurion University of the Negev
P.O.B. 653
Beer-Sheva, 84105
Israel
Abstract.
We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of “inflating” the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Then, under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, an explicit formula for a tensor-factorization of this function, is introduced.
Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research.
today
1. Introduction
The problem of minimal factorization of matrix-valued rational functions of one complex variable has along history; see for instance [1, 2, 6]. Less known seems to be the counterpart of this problem when matrix product is replaced by tensor product. More precisely, we study the following two problems: First, given two rational matrix-valued functions and analytic at infinity, write a realization of the tensor product in terms of realizations of and . Next, given a matrix-valued rational function analytic at infinity, find its representations as where and are rational and analytic at infinity.
To provide some motivation we note the following. Tensor products play an important role in mathematics and quantum mechanics. In the latter case, a first example (see e.g. [4, p. 162]) is the product of two wave functions, each belonging to a given Hilbert space, which belongs to the tensor product of the given Hilbert spaces; see e.g. [8, Proposition 6.2, p. 111] for the latter. Another example is the case of quantum states (positive matrices with trace equal to ; see e.g. [9]). Given two states and , of possibly different sizes, the tensor product is still a state. Note that if , one can recover and uniquely via the formula
[TABLE]
where denotes an orthonormal basis for , and similarly for ,
[TABLE]
where now is an orthonormal basis for . See e.g. [9, eq. (9.2.1) p. 97].
If one starts from an arbitrary state the matrices defined by (1.1) and (1.2) will be states, called marginal states, but their tensor product need not be equal to .
One can consider similar problems in the setting of functions. We focus the discussion on rational functions. If is a -valued rational function and if , formulas (1.1) and (1.2) now define two rational functions and , respectively and -valued, via
[TABLE]
If where is -valued and is -valued, then these equations can be rewritten as
[TABLE]
and so these equations basically solve the tensor factorization problem.
The purpose of this work is in a somewhat different direction; we would like to express both tensor multiplication and tensor factorization of matrix-valued rational functions using state space representations.
In the rest of this section we cite some known results. Let , (the subscript stands for “left” and “right”) be a pair of complex variables, and let , be a pair of , -valued rational functions, respectively. Assume that neither has poles at infinity and denote by , the respective McMillan degrees. Thus, one can write the rational functions and the respective realization as
[TABLE]
Recall that whenever the product is well-defined and its realization is given by111 Strictly speaking, in the references it was formulated for i.e. for (see e.g. [3, Section 2.5])
[TABLE]
in the sense that
[TABLE]
If the sought realization in (1.6) is of McMillan degree
[TABLE]
when minimal (roughly speaking when there is no pole-zero cancelation). We next address ourselves to the tensor product222In matrix theory circles known as the “Kronecker product ”, see e.g. [7, Section 4.2]. of and , resulting in , a -valued rational function. Tensor product of rational functions is discussed in [5, Section 5.2].
So far for known results. In the next section we focus on , the state space realization of . In Section 3 we set the framework for the main result, which is the factorization result presented in Section 4.
2. Realization of a tensor-product of rational functions
We start with technicalities: We denote by boldface characters, “inflated version” of the original ones, i.e.
[TABLE]
We then show that at the expense of “inflating” the dimensions one can replace a tensor product by a usual product.
Proposition 2.1**.**
Let , be a pair of , -valued rational functions, of McMillan degree , , respectively, whose realization is given in Eq. (1.5). Following Eqs. (1.6) and (2.8), one has that,
[TABLE]
In order to go into details we shall repeatedly use the fact, see e.g. [7, Lemma 4.2.10], that for matrices , , , one has that
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We now explicitly compute the tensor product of and ,
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We next separately examine each block
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[TABLE]
[TABLE]
Thus, one can write
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Note that in particular
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The realization of can be compactly written as
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which is indeed in form of (1.6), (2.8). If and there is no pole-zero cancelation, the sought realization in (2.11) is of McMillan degree
[TABLE]
Note now that in a way similar to (1.6), one can factorize the realization in (2.11) as follows,
[TABLE]
We conclude this section by pointing out that Proposition 2.1 can be easily extended to more elaborate cases like
[TABLE]
3. Realization of the inverse of a tensor product of rational functions
For future reference, in this section we examine the realization of the inverse of rational functions of the form studied in the previous section.
We first recall, see e.g. [3, Theorem 2.4], in the realization of the inverse a rational function: Namely if
[TABLE]
is a realization of a square matrix-valued rational function , then whenever is non-singular, is well-defined almost everywhere, and a corresponding realization is given by,
[TABLE]
Next, whenever the above and are so that
[TABLE]
the product is square, and whenever is non-singular333this implies that ., is well-defined almost everywhere, and by combining (1.6) together with (3.13) a corresponding realization is given by
[TABLE]
Similarly, whenever
[TABLE]
the rational function is square and if is non-singular, then , are square, i.e.
[TABLE]
and non-singular, see e.g. [7, Theorem 4.2.15]. Thus, we shall denote hereafter by , the dimensions of , , respectively.
Under these conditions, the -valued rational function, is almost everywhere defined. (2.11), we next compute the realization of ,
[TABLE]
Taking into account the fact that and are square and non-singular, the realization takes the form
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where the boldface entries are given by
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One can conclude that
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and in a way similar to (2.12), one can factorize the above realization as follows,
[TABLE]
4. Tensor-factorization of rational functions
We now address a more challenging question: Given and (assuming that ), under what conditions and how, can it be “tensor-factorized” to some and , namely the following relation holds,
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Note that here, we confine the discussion to a single complex variable, i.e. .
Note also that if (4.1) holds, this is true up to complex scaling i.e.,
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We shall use this degree of freedom in the sequel.
We next recall in the following fact from matrix theory.
Let , be a pair of supporting projections of the space , i.e.
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Such a pair of projections can be obtained by partitioning an arbitrary non-singular as follows.
[TABLE]
By using an isometry-like relation, we next offer a simple way to “deflate” matrix dimensions.
Observation 4.1**.**
Given , denote
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For arbitrary , normalized so that , , one has that
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Indeed, by twice applying (2.10) one obtains,
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We next apply the last observation to the variables here.
Corollary 4.2**.**
For , , normalized so that and , the boldface characters in (2.8) satisfy
[TABLE]
We now return to the problem of “tensor-factorization” in (4.1). We note that in place of in (2.11) and in (3.16), the realization arrays associated with and , are known only up to a coordinate transformation, i.e. there exists, a non-singular matrix namely in (4.2) and (4.3)
[TABLE]
so that the actual realization array is given by
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and
[TABLE]
As in reality, the specific coordinate transformation, in (4.4) and (4.5) is unknown one can conclude that to extract and from (4.1) along with the realization arrays in (4.4), (4.5), additional conditions are needed.
Theorem 4.3**.**
Let be a given square matrix-valued rational function. Assume that
[TABLE]
Let \leavevmode\nobreak\ \footnotesize\left(\begin{array}[]{c|c}\mathbf{A}&\mathbf{B}\\ \hline\cr\mathbf{C}&I\end{array}\right),\leavevmode\nobreak\ see (4.4), and \leavevmode\nobreak\ \footnotesize\left(\begin{array}[]{c|c}\mathbf{A^{\times}}&\mathbf{B^{\times}}\\ \hline\cr\mathbf{C^{\times}}&I\end{array}\right). see (4.5), be realizations of and of , respectively.
Substituting in (4.2), and , assume also that there exists a pair of supporting projection to denoted by and so that
[TABLE]
Following the definition of the projections and , see (4.3) and (4.6), along with Corollary 4.2, for arbitrary , , normalized so that and , we find it convenient to introduce the following related projections444note that and .,
[TABLE]
Then, using (2.11) and (4.4), one can take in (4.1) where,
[TABLE]
Proof : First, recall (see Section 3) that the assumption that is square non-singular, it implies that both and are square non-singular. We shall thus denote the dimensions of and , by and , respectively.
The assumption here that \mathbf{D}=I_{m_{l}m_{r}}\leavevmode\nobreak\ implies (see e.e. [7, Theorem 4.2.12]) that
[TABLE]
As already mentioned after (4.1), to simplify the exposition we shall take .
Next, let in (4.3), (4.4), (4.5) be the same so that the supporting projections are and . Next note that substituting (2.11), (3.16), (4.4) and (4.5) in condition (4.6) yields,
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and thus in the sequel we shall use the two upper relations, i.e.
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We are now ready to recover ,
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Similarly, for
[TABLE]
[TABLE]
∎
Remark 4.4**.**
At first sight, the assumptions in Theorem 4.3 seem very restrictive. For persective recall that to factorize a given rational function to , the assumptions are virtually the same555There they only assume is square non-singular, but then only and are obtained., see [3, Section 2.5]).
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