Block Toeplitz Matrices

Abstract

A Toeplitz matrix is constant along the parallels to the main diagonal. Matrices whose entries in the parallels to the main diagonal form periodic sequences (with the same period N) are referred to as block Toeplitz matrices. Equivalently, A is a block Toeplitz matrix if and only if

$$ A = \left( {\begin{array}{*{20}{c}} {{a_0}}{{a_{ - 1}}}{{a_{ - 2}}} \cdots \\ {{a_1}}{{a_0}}{{a_{ - 1}}} \cdots \\ {{a_2}}{{a_1}}{{a_0}} \cdots \\ \cdots \cdots \cdots \cdots \end{array}} \right) $$

(6.1)

where $\{ a_k \} _{k \in z}$ is a sequence of N × N matrices,${a_k} \in B({C^N})$ for all k ∈ Z.