Skirting the boundaries of being too broad a question, is there any nice way of decomposing a matrix structured as shown below into a product of two or more matrices?

$$ M = \begin{bmatrix} v_1 & v_2 & v_3 & \dots & v_n \\ 0 & v_1 & v_2 & \dots & v_{n-1} \\ 0 & 0 & v_1 & \dots & v_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots &v_1 \\ \end{bmatrix}$$

Is there a special name for such matrices? Where can I read more about them?

  • $\begingroup$ Why even decompose it? What is the motivation? $\endgroup$ – Rodrigo de Azevedo Aug 15 '17 at 10:37
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    $\begingroup$ @RodrigodeAzevedo I was working on a problem that involved diagonalizing circulant matrices (which I recently learnt about) and I was trying to extend the result to other transforms similar to the Fourier transform, which is where I encountered the above matrix. Maybe I should have mentioned the motivation in the question, I was afraid it would not really help with clarifying things. $\endgroup$ – Television Aug 15 '17 at 11:17

It's an upper triangular Toeplitz matrix. Probably "Toeplitz" is the first word to use in your search for further information.

Here, for instance, is a reference about inverting such matrices, and here is an article about factorizing them.

  • $\begingroup$ Thank you, this is what I was looking for. $\endgroup$ – Television Aug 15 '17 at 11:18

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