Let A be an matrix. If is an eigenvalue of and 12nT is an associated eigenvector, use to show that Let k be the largest component of in absolute value. Using part show that Using part show...

Let A be an

matrix.

If

is an eigenvalue of

and


<sub>1

2

n</sub>

^T
is an associated eigenvector, use


to show that

Let

_k
be the largest component of

in absolute value. Using part

show that

Using part

show that each eigenvalue of

satisfies at least one of the inequalities

Argue this means that all the eigenvalues of A lie in the union of the disks



<sub>ii

i</sub>

in the complex plane. This result is one of Gergorin’s disk theorems that are used in perturbation theory for eigenvalues.

Let

Draw the three Gergorin disks and verify that the three eigenvalues lie in their union.