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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) $2 \times 2$ matrices. In particular, I'm interested in solving linear equations of this form. Any reference would be appreciated.

Rodrigo de Azevedo
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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. ain't a mathematician Oct 19 '10 at 23:09
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    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

1 Answers1

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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.

mathreadler
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