Linear Algebra and Its Applications · Problem 7.4.12

The key insight is that a singular matrix $A$ forces any splitting to inherit that singularity through the iteration matrix. When $A = S - T$ is singular, the null space of $A$ becomes a trap: if $Ax = 0$ for some nonzero $x$, then rearranging gives $Sx = Tx$, which means $S^{-1}T$ must map $x$ to itself. This eigenvalue of 1 in the iteration matrix $B = S^{-1}T$ is exactly what kills convergence—the error doesn't shrink, it stays put.

The key insight is that a singular matrix $A$ forces any splitting to inherit that singularity through the iteration matrix. When $A = S - T$ is singular, the null space of $A$ becomes a trap: if $Ax = 0$ for some nonzero $x$, then rearranging gives $Sx = Tx$, which means $S^{-1}T$ must map $x$ to itself. This eigenvalue of 1 in the iteration matrix $B = S^{-1}T$ is exactly what kills convergence—the error doesn't shrink, it stays put.