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?
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.