A short note on Multilevel Toeplitz Matrices
Lei Cao, Selcuk Koyuncu

TL;DR
This paper extends the known result that one-level Toeplitz matrices are unitarily similar to complex symmetric matrices to multilevel Toeplitz matrices, providing a construction method and exploring related classes.
Contribution
It introduces a method to construct unitary matrices for multilevel Toeplitz matrices, generalizing previous results and identifying classes of matrices related to Toeplitz structures.
Findings
Multilevel Toeplitz matrices are unitarily similar to complex symmetric matrices.
A tensor product-based method constructs the necessary unitary matrices.
A class of matrices similar to multilevel Toeplitz matrices is characterized.
Abstract
Chien, Liu, Nakazato, and Tam proved that all n by n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper, we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.
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Taxonomy
TopicsHolomorphic and Operator Theory · Advanced Algebra and Geometry · Finite Group Theory Research
A Short Note on Multilevel Toeplitz Matrices
Lei Cao 11footnotemark: 1
Department of Mathematics, Halmos College, Nova Southeastern University, FL 33314
Selcuk Koyuncu
Department of Mathematics, University of North Georgia, GA 30566
Abstract
Chien, Liu, Nakazato and Tam proved that all classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some -level Toeplitz matrices.
keywords:
Multilevel Toeplitz matrix; Unitary similarity; Complex symmetric matrices
MSC:
[2010] 15B05; 15A15
††journal: Linear Algebra and Its Applicationsmytitlenotemytitlenotefootnotetext: Fully documented templates are available in the elsarticle package on CTAN.
1 Introduction
Although every complex square matrix is unitarily similar to a complex symmetric matrix (see Theorem 4.4.24, [5]), it is known that not every matrix is unitarily similar to a complex symmetric matrix when (See [4]). Some characterizations of matrices unitarily equivalent to a complex symmetric matrix (UECSM) were given by [1] and [3]. Very recently, a constructive proof that every Toeplitz matrix is unitarily similar to a complex symmetric matrix was given in [2] in which the unitary matrices turning all Toeplitz matrices to complex symmetric matrices was given explicitly. An interesting fact was that the unitary matrices only depend on the parity of the size.
Multilevel Toeplitz matrices arise naturally in multidimensional Fourier analysis when a periodic multivariable real function is considered [6]. In this paper, we show that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. Along the line in [2], a constructive proof is given. One can take tensor product of the unitary matrices defined in [2] and identity matrices appropriately to construct the unitary matrix turning any multilevel Toeplitz matrix to a complex symmetric matrix which only depends on the parity of the size of each level. In section 4, we provide two examples of constructing the unitary transition matrices of a 2-level Toeplitz matrix and a 3-level Toeplitz matrix to illustrate our main results in section 3. The converse is considered in Section 5, in which we give the necessary and sufficient condition for a complex symmetric matrix similar to a -level Toeplitz matrix under the unitary transformation given in Section 3.
2 Preliminary and Notations
A classical 1-level matrix is called Toeplitz if it has constant entries along its diagonals, i.e, if it is of the form
[TABLE]
A -level Toeplitz matrix, denoted by has Toeplitz structure on each level and corresponds to a -variate generating function.
For an integer a -level Toeplitz matrix of size where and for is a block Toeplitz matrix of the form
[TABLE]
where each block is itself a -level Toeplitz matrix of size For instance if p=2, we have the following two-level Toeplitz matrix with Toeplitz blocks
[TABLE]
where and are classical -level Toeplitz matrices.
More generally, let For let with Denote for Denote
[TABLE]
where Then a -level Toeplitz matrix, is of size and denoted by
[TABLE]
where the th block of is the -level Toeplitz matrix, of size for Note that -level Toeplitz matrix is a regular Toeplitz matrix. Using the notation of -level Toeplitz matrices, the main result in [2] is stated as the following theorems.
Theorem 2.1
(Theorem 3.3 [2]) Every -level Toeplitz matrix is unitarily similar to a symmetric matrix. Moreover, the following by even and odd unitary matrices uniformly turn all Toeplitz matrices with even sizes and odd sizes into symmetric matrices respectively via similarity:
- (i)
when with
[TABLE] 2. (ii)
when with
[TABLE]
Let be the matrix with all elements zero except the elements on the anti diagonal which are all s. That is,
[TABLE]
Then a Topeplitz matrix with any size can be unitarily turned into a symmetric matrix by the matrix
[TABLE]
which is clearly unitary.
Theorem 2.2
(Theorem 3.1 [2]) Every Toeplitz matrix is unitarily similar to a symmetric matrix via the unitary matrix
[TABLE]
More specifically,
[TABLE]
3 Multilevel Unitary Symmetrization
Denote an unitary matrix and if is even, is defined by (2); if is odd, is defined by (3).
Theorem 3.1
Let be a -level Toeplitz matrix of size . Then there exists a unitary matrix of size such that
[TABLE]
is symmetric and the unitary transition matrix is
[TABLE]
where
[TABLE]
for
Proof 3.2
We prove it by mathematical induction on .
For it is true due to Theorem 3.3 in [2].
Assume the result is true for meaning that there exists a unitary matrix of size such that
[TABLE]
is symmetric for any -level Toeplitz matrix with size
That is, any -level Toeplitz matrix is unitarily similar to a symmetric matrix via This implies the following
[TABLE]
is symmetric.
Let us prove the result for case
Consider a -level Toeplitz matrix with size
[TABLE]
where all blocks are -level Toeplitz matrices of size and note that Next we define
[TABLE]
Let Then
[TABLE]
By induction hypothesises, is symmetric. Denote by then
[TABLE]
where is symmetric for
Let
[TABLE]
that is,
[TABLE]
if is even;
[TABLE]
if is odd.
It suffices to show that is symmetric. Let
Suppose is even, that is for some integer Then
[TABLE]
and
[TABLE]
which gives us that
[TABLE]
Denote
[TABLE]
where and have the same size and let be the indices. Then we get,
- (i)
For and
[TABLE] 2. (ii)
For and
[TABLE] 3. (iii)
For and
[TABLE] 4. (iv)
For and
[TABLE]
First note that (4) and (7) are the same due to the Toeplitz structure of If we switch and in (4) or (7), we have
[TABLE]
which is equal to (4) and (7) meaning that both and are symmetric. If we switch and in (5), we have
[TABLE]
equal to (6) which shows that and Hence
[TABLE]
Thus is symmetric.
- 2.
*Suppose is odd. Then we can write for some integer Let Similarly to the case for even, one can show for and In addition, straightforward calculation yields the **th row and the *th column as follows
[TABLE]
Hence is symmetric.
We also generalize Theorem 2.2, in which one does not need to consider the parity of the size. We denote
[TABLE]
Theorem 3.3
Let be a -level Toeplitz matrix of size Then there exists a unitary matrix such that is symmetric, where
[TABLE]
and
[TABLE]
for
Proof 3.4
The proof will be omitted since it is similar to Theorem 3.1.
4 Examples
Here are two examples to illustrate the constructions of the transition matrices given by Theorem 3.1 and Theorem 3.3 respectively .
Example 1
Let
[TABLE]
a -level Toeplitz matrix of size where and By Theorem 3.1,
[TABLE]
Then
[TABLE]
and
[TABLE]
So
[TABLE]
in which each block is symmetrized, that is the first level is symmetrized, and
[TABLE]
which is symmetric. The transition unitary matrix is given by
[TABLE]
One may use Theorem 3.3 as well. To construct the transition matrix, we construct and as the following:
[TABLE]
Then
[TABLE]
and
[TABLE]
So
[TABLE]
where the transition unitary matrix is given by
[TABLE]
Example 2
Let
[TABLE]
a -level Toeplitz matrix of size where By Theorem 3.1,
[TABLE]
and hence
[TABLE]
[TABLE]
and
[TABLE]
respectively. So the transition unitary matrix is
[TABLE]
and one can check that
[TABLE]
symmetric.
Now we are using Theorem 3.3 to symmetrize the same -level Toeplitz matrix.
[TABLE]
[TABLE]
and
[TABLE]
respectively. So the transition unitary matrix is
[TABLE]
and one can check that
[TABLE]
is symmetric and note that the resulting symmetric matrices are not necessarily the same.
5 Symmetric matrices that are unitarily similar to Toeplitz matrices
Let be an -level Toeplitz matrix. According to Theorem 3.1, there exists a unitary matrix such that is a symmetric matrix. However, the converse is not true, i.e., not every complex symmetric matrix is unitarily similar to a (multilevel) Toeplitz matrix (see Section 5 and Section 6 in [2]). Denote the set of all complex symmetric matrices. In this section, we provide the necessary and sufficient condition under which a matrix in is similar to a -level Toeplitz matrix under the unitary transformation given in Section 3.
Let Let be a positive integer less than or equal to Then can be written as
[TABLE]
where and each is a matrix for For each is called a -level block of and is said to have -level constant anti-diagonals if each has a constant anti-diagonal. having constant anti-diagonals at each level means that has -level constant anti-diagonals for all
Example 3
Let and
[TABLE]
* having -level constant anti-diagonals means that*
[TABLE]
* having -level constant anti-diagonals means that*
[TABLE]
* having constant diagonals at each level means both (8) and (9).*
Given a positive integer Let Then let
[TABLE]
and
[TABLE]
for Denote
[TABLE]
Lemma 5.1
Let be a -level Toeplitz matrix. Let be the unitary matrix defined by (10). Then the complex symmetric matrix
[TABLE]
has constant anti-diagonals at each level.
Proof 5.2
We use induction on
Base case: When has a constant anti-diagonal due to the symmetry of
Inductive assumption: Suppose it is true for That is the complex symmetric matrix has constant anti-diagonals on each level.
Inductive step: We need to show for the complex symmetric matrix has constant anti-diagonals on each level.
First note that
[TABLE]
where and are -level Toeplitz matrices. According to the inductive assumption, there exists a unitary matrix such that all
[TABLE]
are complex symmetric matrices with constant anti-diagonals at each level. That is,
[TABLE]
Let
[TABLE]
where is the identity matrix. Then
[TABLE]
which has constant anti-diagonals at each level.
Lemma 5.3
Let be a complex symmetric matrix. If has constant anti diagonals for each level, then is a -level Toeplitz matrix.
Proof 5.4
We use induction on
Base case: For Let Then
[TABLE]
a -level Toeplitz matrix.
Inductive assumption: Suppose it is true for That is, for a complex symmetric matrix with constant anti-diagonals at each level, is an -level Toeplitz matrix.
Inductive step: We need to show for if is a complex symmetric matrix with constant anti-diagonals at each level, then is a -level Toeplitz matrix.
Note that
[TABLE]
where and are complex symmetric matrices with constant anti-diagonals at each level. According to the inductive assumption, there exists a unitary matrix such that all
[TABLE]
are -level Toeplitz matrices. That is,
[TABLE]
We define as
[TABLE]
where is the identity matrix. Then
[TABLE]
is a -level Toeplitz matrix.
Combine Lemma 5.1 and Lemma 5.3, we have the following theorem.
Theorem 5.5
Let be a complex symmetric matrix. There exists a -level Toeplitz matrix such that
[TABLE]
if and only if has constant anti-diagonals at each level.
Acknowledgements
We would like to thank Drs.Banani Dhar and Sarita Nemani for insightful discussions.
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