So I have a bunch of linear equations $Ax=y : A \in R^{m,m}, y \in R^{m}$. Note that $A$ is a square matrix. The question is if I can decompose $A$ as $$A = D + uv^T$$ where $D$ is diagonal, invertible, and $u,v \in R^N$. Would I be able to use the structure of this matrix to my advantage in solving the equation. My initial hunch was that it would somehow make #QR$ decomposition easier but I'm having trouble with this approach. Could someone point me in the right direction? I don't want an answer but just a simple clue.
2026-04-17 22:32:50.1776465170
Exploiting structure of linear equations to solve them
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You can use the Sherman-Morrison formula:
$$(A+uv^T)^{-1} = A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}.$$
In your case, since $D$ is diagonal and invertible, the formula boils down to something quite simple. (Recall, the inverse of another diagonal matrix is a diagonal matrix with the reciprocals of the diagonal elements)