does anyone know the name of this matrix or if it has any special characteristics or how to calculate its inverse efficiently e.g. in a closed-form?
[ \begin{array}{llllll} a_{11} & a_{12}& a_{13} & a_{14} & \dots & a_{1L} \\ q_1 & a_{22} & a_{23} & a_{23} & \dots & a_{2L} \\ q_2 & q_1 & a_{33} & a_{34} & \dots & a_{3L} \\ q_3 & q_2 & q_1 & a_{44} & \dots & a_{3L} \\ q_4 & q_3 & q_2 & q_1 & \dots & a_{3L} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ q_L & q_{L-1} & q_{L-2} & q_{L-3} & \dots & a_{LL}\\ \end{array}]
hint: the lower triangle consists of diagonals with the same value.
This matrix is not always invertible, e.g. if $a_{i,j} = 0$ for every $i,j$ or if $a_{i,j}=q_i=c$ for every $i,j$.
Anyway you may recognize the structure of a Toeplitz matrix on the lower triangle.