A note on multilevel Toeplitz matrices


Chien, Liu, Nakazato and Tam proved that all n × 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.


Introduction
Although every complex square matrix is similar to a complex symmetric matrix (see Theorem 4.4.24, [5]), it is known that not every n × n matrix is unitarily similar to a complex symmetric matrix when n ≥ (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 n × n 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 de ned 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 su cient condition for a p × p complex symmetric matrix similar to a p-level Toeplitz matrix under the unitary transformation given in Section 3.

Preliminary and Notations
A classical 1-level matrix Tn ∈ C n×n is called Toeplitz if it has constant entries along its diagonals, i.e, if it is of the form A p-level Toeplitz matrix, denoted by T (p) , has Toeplitz structure on each level and corresponds to a pvariate generating function.
For an integer p ≥ , a p-level Toeplitz matrix of size (n n n n · · · np) × (n n n · · · np) where n = and n i ∈ N for i = , , . . . , p, is a block Toeplitz matrix of the form is itself a (p − )-level Toeplitz matrix of size (n · n · · · n p− ) × (n · n · · · n p− ). For instance if p=2, we have the following two-level Toeplitz matrix with Toeplitz blocks where T ( ) = matrices.
More generally, let p ∈ N. For ≤ i ≤ p, let n i ∈ N with n = . Denote s k = k i= n i for k = , , . . . , p. Denote where t i−j ∈ C. Then a p-level Toeplitz matrix, T (p) is of size sp × sp and denoted by where the (i, j)th block of T (p) is the (p − )-level Toeplitz matrix, T (p− ) i−j , of size s p− for |i − j| ≤ np − . Note that -level Toeplitz matrix T ( ) is a regular Toeplitz matrix. Denote i = √ − . Using the notation of p-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 T ∈ Cn×n is unitarily similar to a symmetric matrix. Moreover, the following n × n even and odd unitary matrices uniformly turn all Toeplitz matrices with even sizes and odd sizes into symmetric matrices respectively via similarity: (a) when n = m with m ≥ , Let Jn be the n × n matrix with all elements zero except the elements on the anti diagonal which are all s. That is, Then a Toeplitz matrix with any size can be unitarily turned into a symmetric matrix by the matrix which is clearly unitary. More speci cally,

Multilevel Unitary Symmetrization
Denote U(n) an n × n unitary matrix and if n is even, U(n) is de ned by (2); if n is odd, U(n) is de ned by (3).
Proof. We prove it by mathematical induction on p.
For p = , it is true due to Theorem 3.3 in [2]. Assume the result is true for k meaning that there exists a unitary matrixŨ of size s k × s k such that is symmetric for any k-level Toeplitz matrix with size s k × s k . That is, any k-level Toeplitz matrix T (k) is unitarily similar to a symmetric matrix viaŨ = U · · · U k . This implies the following where all blocks T (k) i−j are k-level Toeplitz matrices of size s k × s k and note that s k = k i= n i . Next we de nê whereS t is symmetric for t = −n k+ + , −n k+ + , . . . , − , , , . . . , n k+ − . Let U k+ = U(n k+ ) ⊗ Is k , that is, if n k+ is even; Suppose n k+ is even, that is n k+ = t for some integer t. Then where Z , Z , Z and Z have the same size and let ≤ p, q ≤ n k+ be the indices. Then we get, Unauthenticated Download Date | 10/12/19 1:19 AM (c) For t + ≤ p ≤ t and ≤ q ≤ t, First note that (4) and (7) are the same due to the Toeplitz structure of S. If we switch p and q in (4) or (7), we haveS which is equal to (4) and (7) meaning that both Z and Z are symmetric. If we switch p and q in (5), we have equal to (6) which shows that Z = Z t and Z t = Z . Hence Thus S is symmetric.
Suppose n k+ is odd. Then we can write n k+ = t + for some integer t. Let S = V * S V . Similarly to the case for even, one can show Spq = Sqp for p = , , . . . , t, t + , . . . , t + and q = , , . . . , t, t + , . . . , t + . In addition, straightforward calculation yields the (t + )th row and the (t + )th column as follows We also generalize Theorem 2.2, in which one does not need to consider the parity of the size. We denote V(n) = √ (In + iJn). Proof. The proof will be omitted since it is similar to Theorem 3.1.

Theorem 3.2. Let T (p) be a p-level
a -level Toeplitz matrix of size × , where n = and n = . By Theorem 3.1, in which each block is symmetrized, that is the rst level is symmetrized, and which is symmetric. The transition unitary matrix U is given by One may use Theorem 3.2 as well. To construct the transition matrix, we construct V(n ) and V(n ) as the following: a -level Toeplitz matrix of size × , where n = n = n = . By Theorem 3.1,

respectively. So the transition unitary matrix is
and one can check that symmetric. Now we are using Theorem 3.2 to symmetrize the same -level Toeplitz matrix.
respectively. So the transition unitary matrix is and one can check that is symmetric and note that the resulting symmetric matrices are not necessarily the same.

Symmetric matrices that are unitarily similar to Toeplitz matrices
Let Tn be an n × n p-level Toeplitz matrix. According to Theorem 3.1, there exists a unitary matrix U, such that UTn U * 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 S p the set of all p × p complex symmetric matrices. In this section, we provide the necessary and su cient condition under which a matrix in S p is similar to a p × p p-level Toeplitz matrix under the unitary transformation given in Section 3. Let S ∈ S p . Let q be a positive integer less than or equal to p. Then S can be written as has constant anti-diagonals at each level.
Proof. We use induction on p.
Base case: When p = , S ( ) has a constant anti-diagonal due to the symmetry of S ( ) .
Inductive assumption: Suppose it is true for p = m. That is the m × m complex symmetric matrix S (m) = (U (m) ) * T (m) U (m) has constant anti-diagonals on each level.
Inductive step: We need to show for p = m + , the m+ × m+ complex symmetric matrix S m+ = (U (m+ ) ) * T (m+ ) U (m+ ) has constant anti-diagonals on each level. Inductive assumption: Suppose it is true for p = m. That is, for a m × m complex symmetric matrix S (m) with constant anti-diagonals at each level, U (m) S (m) (U (m) ) * is an m-level Toeplitz matrix.
Note that if and only if S has constant anti-diagonals at each level.