Linear Algebra Done Right · Problem 5D.22

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · gershgorin-disk-theorem

Problem

Suppose

T \in \mathcal{L}(V)

and

A

is an

n

-by-

n

matrix that is the matrix of

T

with respect to some basis of

V

. Prove that if

|A_{j,j}| > \sum_{\substack{k=1\\k\neq j}}^{n} |A_{j,k}|

for each

j \in \{1, \ldots, n\}

, then

T

is invertible. This exercise states that if the diagonal entries of the matrix of

T

are large compared to the nondiagonal entries, then

T

is invertible.