Chapter 2.5, Problem 5E

Let $\lambda$ be an eigenvalue of an $n\times n$ matrix A. (a) Prove the Gershgorin Circle Theorem: There is a diagonal entry $A_{mm}$ such that $\left|A_{mm}-\lambda \right|\le {\displaystyle {\sum }_{j\ne m}\left|A_{mj}\right|}$ . (Hint: Begin with an eigenvector V such that ${\Vert v\Vert }_{\infty }=1$, as in the proof of Theorem 2.10.) (b) Prove that a strictly diagonally dominant matrix cannot have a zero eigenv&ue. This is an alternative proof of part (1) of Theorem 2.10.