Inverse of particular lower triangular matrix

Question

I have an $n \times n$ lower triangular matrix $A$ where

$$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$

$$A_{ii}=1, \quad 1 \leq i \leq n,$$

and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $k < n$. I need to study $A^{-1}$.

Since matrix $A$ is formed from (roughly) $n k$ elements, but contains (roughly) $n^2$, there could be some special properties. Any input on the structure would be useful. Currently, I use a simple Neumann series to study the behavior of the inverse, but with limited results.

Answer 1

One observation:

$$A = I+L,$$ where $L$ is a lower triangular matrix with $0$ in the diagonals. This matrix $L$ can be seen to satisfy $L^n=0$, and $L^j\ne 0,\ 1\le j\le n-1$. Thus, one can write $$A^{-1}=(I+L)^{-1}=\sum_{j=0}^{n-1}(-1)^jL^j$$

Answer 2

$A$ is $(k,0)$-semi-separable, a property that is preserved through inversion. This means, in particular, that every submatrix contained in the strictly lower diagonal part of $A^{-1}$ (as well as $A$) has rank at most $k$.

There are around algorithms to perform linear algebra computations on these matrices efficiently. See "Matrix Computations & Semiseparable Matrices", parts I and II, by Vandebril R., Van Barel M. and Mastronardi N. "Efficiently" usually means $O(n^2)$, so I am not sure how much of that theory will be relevant for a lower triangular $A$.

I'm not sure that this is the kind of properties that you were looking for, but in any case it may be of help.