# Block Toeplitz Matrix

What is the name of the following matrix:

$$\begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & a & b\\ 0& c& d& \\ b & 0 & a \\ d & 0 & c\end{pmatrix}\$$

It looks like a Block Toeplitz matrix, but usually one defines those by full shifts by (in this case) 2x2 matrices. In particular i'm interested in solving linear equations of this form. Any reference would be appreciated.

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If $A$ denotes your matrix, I assume that you're looking for solutions to the matrix equation $Ax = y$. Do you have any restrictions on the values of $a, b, c, d$ such that you are guaranteed a solution? (In general, systems with six equations and only three unknowns are not likely to have a solution.) – Mike Spivey Oct 19 '10 at 20:26
You're definitely going to have to solve your problem the least-squares way (or in general ,minimize with respect to whatever other norm that you're using, but that's more complicated). At first glance I can't see how to modify QR or SVD to exploit your system's structure. – J. M. Oct 19 '10 at 23:09
it doesn't just "look" like a block-Toeplitz matrix, it IS one! Simply define the two 2x1 blocks: [a;c] and [b;d]... – Laurent Lessard Oct 20 '10 at 17:50

You could call it block circulant even, which is more restrictive than Toeplitz. A block matrix does not need every block to be the same size. As Laurent wrote above: Just consider the blocks to be $2 \times 1$ matrices instead of $2 \times 2$ and you will be fine.