Linear Algebra and Its Applications · Problem 7.4.4

The key insight is the Gershgorin Circle Theorem: each eigenvalue of $A$ must lie inside at least one circle centered at a diagonal entry $a_{ii}$ with radius $r_i$ (the sum of absolute values of off-diagonal entries in that row). If you can show that zero lies outside all these circles, then zero cannot be an eigenvalue—which means $A$ is nonsingular. Diagonal dominance is precisely the condition that guarantees this separation.

The key insight is the Gershgorin Circle Theorem: each eigenvalue of $A$ must lie inside at least one circle centered at a diagonal entry $a_{ii}$ with radius $r_i$ (the sum of absolute values of off-diagonal entries in that row). If you can show that zero lies outside all these circles, then zero cannot be an eigenvalue—which means $A$ is nonsingular. Diagonal dominance is precisely the condition that guarantees this separation.