Suppose the eigenvalues of a symmetric matrix A satisfy λ 1 > λ 2 ≥ ··· ≥ λ n −1 > λ n > 0. To calculate λ 1 , the power method is going to be applied to the shifted matrix B = A − I . (a) What...

Suppose the eigenvalues of a symmetric matrix
A
satisfy λ₁
> λ₂
≥ ··· ≥ λ
<sub>n

−1</sub>
> λ
_n

> 0. To calculate λ₁, the power method is going to be applied to the shifted matrix
B
=
A
−

I.

(a) What condition(s) must be imposed on
 so the method will converge to λ₁.

(b) Explain why the choice
 = (λ₂
+ λ
_n
)/2 will result in the fastest convergence of the power method.