Let $S,T\in\mathbb{R}^{n\times n}$ be 2 upper triangular matrixes, and $ST-\lambda I$ be nonsingular where $I$ denotes the identity matrix of order $n$. Solve the system of linear equations in complexity $O(n^2)$ for $b\in\mathbb{R}^n$: $$(ST-\lambda I)x=b$$
It's known that equations like $Uy=b$ for upper triangular $U$ can be solved in complexity $O(n^2)$ by back substitution method. I tried to rewrite $ST-\lambda I$ as the product of 2 matrixes, but it failed. Now I get stuck on this puzzling problem. Appreciated for any suggestions.