SOME RESULTS CONCERNING BLOCK TOEPLITZ MATRICES

. Given A, B, C, and D, block Toeplitz matrices, we will prove the necessary and suﬃcient condition for AB − CD = 0, and AB − CD to be a block Toeplitz matrix. In addition, with respect to change of basis, the characterization of normal block Toeplitz matrices with entries from the algebra of diagonal matrices is also obtained.


Introduction
A scalar Toeplitz matrix is an n × n matrix with the following structure: . . a n−1 a −1 a 0 a 1 . . . a n−2 a −2 a −1 a 0 . . . a n−3 . . . The entries depend upon the difference i − j and hence they are constant down all the diagonals. The subject is many decades old; among monograph dedicated to the subject are [7,12] and [26]. These matrices are ubiquitous and arise naturally in several theoretical and applicative fields. In particular, mathematical modelling of all the problems where to some extent the shift invariance appears in terms of space or of time. This shift invariance is contemplated in the structure of the matrix itself where a shift on south-eastern leaves the matrix fixed.
For two dimensional problems the structure of Toeplitz may also occur block wise.
1.1. Problems modelled by (block) Toeplitz matrices. There is a natural relation between problems involving Toeplitz structures and power series (even with Laurent series in case of infinite Toeplitz structure) which allows one to shift from algorithms for Toeplitz computations to algorithms for power series computations and vice versa. Fast Fourier Transform in this way becomes a principal tool for all the computations involving Toeplitz-like matrices. In the last decades a large amount of research has been concentrated on the analysis of algorithms for Toeplitz matrices. In particular, there are two specific problems: (i) solving linear system involving Toeplitz matrix of dimension d, where the matrix is generated by a real function; see [1,4,6].
Restoring a blurred image in this way, is reduced to solving a block banded block Toeplitz systems with banded Toeplitz blocks. Such type of basic problems appears in many forms in image processing [9,21,22,24].
1.4. Block Toeplitz matrices in signal processing. Signal processing plays a vital role in multiple industries and it is the technology of future. We now give two interesting examples of block Toeplitz matrices that are frequently used in Signal Processing as well as in Communications and Information Theory; see [6,14,25].
(i) Let {η d } d∈Z be a sequence in C 1 ⊗ C , i.e., η d ∈ C 1 ⊗ C for every d ∈ Z. If j, k ∈ Z with j ≤ k, then Then consider a discrete time causal finite impulse response (FIR) multiple input multiple output (MIMO) filter, that is, a filter given as where the filters taps A −k , with 0 ≤ k ≤ r are m × n block matrices, and the input {η d } d∈Z and the output {ζ d } d∈Z of the filters satisfy that η d ∈ C 1 ⊗ C and ζ d ∈ C 1 ⊗ C for every d ∈ Z. It follows then from (1.1) that Thus, representations of discrete time causal FIR MIMO filters in terms of matrix are block Toeplitz.
(ii) We now consider a vector wide sense stationary (WSS) process. Suppose that η n be a random vector of dimension d for every n ∈ Z. Suppose that the d dimensional multivariate random processes {η n } n∈Z has constant mean, i.e., and if it obeys Thus, the corelation matrix E(η n:1 η * n:1 ) of the random of vector η n:1 is an n × n block Toeplitz matrix with d × d blocks for every n ∈ N.
There is an immense literature concerning scalar Toeplitz matrices; on the other side, research related to block Toeplitz matrices is rather sparse. In this paper, we intend to add some remarks and results to the latter. Our starting point is the paper of Gu and Patton [8], which provides various properties of usual scalar Toeplitz and Hankel matrices; most of the results therein refer to products of these structured matrices, which, in general, are no more structured.
The area of block Toeplitz matrices is less studied, one of the reason being the new difficulties that appear with respect to the scalar case. Besides its theoretical interests, the subject is also important to multivariate control theory. We refer the reader to [2,5,17,19,23] and further references thereafter for a detailed study about block Toeplitz matrices.
In [8], the authors have proved a variety of algebraic results about scalar Toeplitz matrices. Given Toeplitz matrices A, B, C, and D, their main result determines if the matrix AB − CD is Toeplitz. The necessary and sufficient condition is a rank two matrix equation involving tensor products of the vectors defining A, B, C, and D. They have also proved the necessary and sufficient condition for AB − CD = 0. In addition to that, they have also completely characterized normal Toeplitz matrices. The characterization of normal Toeplitz matrices has been discussed in [3,11,13].
The purpose of the present paper is to generalize some of the results of [8] concerning the product of block Toeplitz matrices by way of introducing the special structure of the displacement matrix of a block Toeplitz matrix.
The outline of the paper is as follows: Notations and some basic facts about displacement matrices are presented in section 2. Then as we will be interested in block Toeplitz matrices, some basic properties concerning the product of these matrices are derived in section 3. The section 4 is devoted to the study of normal block Toeplitz matrices with entries from the algebra of diagonal matrices.

Preliminaries
As usual N, C, and Z denote the set of natural numbers, complex numbers and integers respectively. We symbolize by M n the algebra of n × n matrices and by D d the algebra of d × d diagonal matrices with entries from C . Throughout in this paper, we label the indices from 0 to n − 1; so A ∈ M n is written A = (a i,j ) n−1 i,j=0 with a i,j ∈ C. Then T n ⊂ M n is the space of scalar Toeplitz matrices A = (a i−j ) n−1 i,j=0 . We will mostly be interested in block matrices, i.e., matrices whose elements are not necessarily scalars, but elements in M d . Thus a block Toeplitz matrix is actually an nd × nd matrix, but which has been decomposed in n 2 blocks of dimension d, and these blocks are constant parallel to the main diagonal. In the sequel we will use the following notations: • M n ⊗M d is the collection of n×n block matrices whose entries all belong to M d ; • T n ⊗M d is the collection of n×n block Toeplitz matrices whose entries all belong to M d ; • D n ⊗ M d is the collection of n × n diagonal block Toeplitz matrices whose entries all belong to M d ; • C 1 ⊗M d is the collection of all n×1 block matrices whose entries all belong to M d ; • R 1 ⊗ M d is the collection of all 1 × n block matrices whose entries all belong to M d .
For block diagonal matrices we will use the notation In most cases, it will suffice to consider block Toeplitz matrices with zero diagonals. In other casesÃ +Ã 0 will describe the most general block Toeplitz matrix , whereÃ is a block Toeplitz matrix with 0 on the main diagonal andÃ 0 is the diagonal block Toeplitz matrix. We will use this matrix to determine whether the difference of the matrix products is Toeplitz. For other kinds of displacement matrices see [10] and [15] . We denote the vector then its adjoint is the vector P * The following lemma is quite useful for proving our main results of section 3.
Thus if want to show that M = 0, then it will sufficient to show that △(M) = 0 .

Block Toeplitz product
In this section, we will generalized some important results of [8]. The following Lemma describes the necessary and a sufficient condition for a block matrix A to be a block Toeplitz matrix. Proof. Suppose that A = (A i−j ) n−1 i,j=0 ∈ T n ⊗ M d . Since, the displacement matrix for A is defined as △(A) = A − SAS * . Then simple computation yields that

Compairing corresponding entries yields
A is a block Toeplitz matrix.
In the rest of this section, if A = (A i−j ) n−1 i,j=0 is any block Toeplitz matrix then for simplification we write A =Ã+Ã 0 , whereÃ ∈ T n ⊗M d , with 0 on the main diagonal andÃ 0 ∈ D n ⊗ M d . The next lemma is the main technical result of this section.
Proof. We have SinceB 0 ∈ D n ⊗ M d , thenÃB 0 ∈ T n ⊗ M d , therefore by Lemma 3.1, there exist U, U ′ ∈ C 1 ⊗ M d such that △(ÃB 0 ) = UP + + P * + U ′ * . Similarly △(Ã 0B ) = V P + + P * + V ′ * , and △(Ã 0B0 ) = W P + + P * + W ′ * , with V, V ′ , W, W ′ ∈ C 1 ⊗ M d . Therefore We have entry at the position (i, j) ofÃB is Using the formula (3.2) and definition of S, one obtains (3.3) Therefore . Combining (3.1) and (3.4) yields The next Theorem is the most important result of this paper. It gives answer to the question that when the product AB − CD of block Toeplitz matrices A, B, C, and D is in T n ⊗ M d .

Normal block Toeplitz matrices
The general problem of characterizing normal block Toeplitz matrices with entries from M d is an open problem. As a first step, we take entries of block Toeplitz matrices from D d , then we will obtain the characterization of normal block Toeplitz matrices. We start with the following result, which is Lemma 5.1 of [17] . i,j=0 ∈ T n ⊗ D d . Then there is a change of basis that brings A into the following form where for every k = 1, 2, · · · d, A ′ k ∈ T d . The next result from [8] characterized normal Toeplitz matrices among all Toeplitz matrices.
Theorem 4.2. Let A = (a i−j ) n−1 i,j=0 ∈ T n , then A is normal if and only if either a = λb for some |λ| = 1 or a = λb for some |λ| = 1.
The following result is the main result of this section. Although, it is a trivial implications of Lemma 4.1 but it will provide a basis for future research concerning the characterization of normal block Toeplitz matrices with entries from M d . Theorem 4.3. Let A ∈ T n ⊗ D d , then A is normal if and only if there exist scalars λ 1 , λ 2 , · · · λ d with |λ k | = 1, such that for every k = 1, 2, · · · d, either a k = λ kb or a k = λ kbk .
Proof. Suppose A ∈ T n ⊗ D d , then by Lemma 4.1, A has the form where for every k = 1, 2, · · · d, A ′ k = (a r−s,k ) d r,s=0 . Since A ′ ∈ D n ⊗ M d , then AA * = A * A if and only if A ′ k A ′ * k = A ′ * k A ′ k . By Theorem 4.2, each A ′ k is normal if and only if there exist scalars λ 1 , λ 2 , · · · λ d , with |λ k | = 1, such that either a k = λ kb or a k = λ kbk , where a k =         , k = 1, 2, · · · d. The proof is therefore complete.