This problem considers the following × tri-diagonal matrix: It is possible to show that the eigenvalues of this matrix are Also, assume that ≥ 3. (a) What is ||A||∞, and what is ||A||1? (b) In the...

This problem considers the following

×

tri-diagonal matrix:

It is possible to show that the eigenvalues of this matrix are

Also, assume that

≥ 3.

(a) What is ||A||
_∞
, and what is ||A||₁?

(b) In the case that
A
is symmetric (so,
=
), what inequality must be satisfied to guarantee that
A
is strictly diagonal dominant?

(c) Assuming the matrix is symmetric and

is positive with 2|| ≤
, explain why
A
is positive definite.