Characterize the family of all $n\times n$ nonsingular matrices $A$ for which one step of the Gauss-Seidel algorithm solves $Ax=b$, starting at

Question

Characterize the family of all $n\times n$ nonsingular matrices $A$ for which one step of the Gauss-Seidel algorithm solves $Ax=b$, starting at the vector $x=0$

I know that if $A$ is strictly diagonally dominant, then the Gauss-seidel method converges for any starting vector $x^{(0)}$. How can I use this to find how $A$ should be? Thank you very much.

Answer 1

If $A = L + U$, and our initial vector is $0$, what is the next approximation of a solution that Gauss-Seidel computes?

That next approximation must be $b$ (that's what "one step ... solves" means). So, what does that tell you about $L$, $U$ and $A$?

You need to at least try these before someone here is going to give you anything like an answer.