Solving for an lower triangular matrix in a matrix inverse

Question

Let $a$ and $b$ be two strictly positive vectors, and consider the following equation $$ a^T=b^T(I-X)^{-1} $$ where $X$ is a lower triangular matrix and $I$ is the identity. Is there a unique matrix $X$ that solves this equation? If so, is there a closed-form expression for $X$? I think the answer are yes but I struggle for a general result in $n$ dimensions.

Answer 1

$$ {\bf a}^\top ({\bf I}_n - {\bf X}) = {\bf b}^\top $$

can be rewritten as follows

$$ {\bf a}^\top {\bf X} = {\bf a}^\top - {\bf b}^\top $$

Half-vectorizing both sides, we obtain an underdetermined system of $n$ linear equations in $\binom{n+1}{2}$ unknowns, as follows

$$ \left( {\bf I}_n \otimes {\bf a}^\top \right) {\bf D}_n \mbox{vech} ({\bf X}) = {\bf a} - {\bf b} $$

where $\otimes$ denotes the Kronecker product and ${\bf D}_n$ is the duplication matrix.