Suppose the n n matrix A has eigenvalues 1, . . . , n ordered by

Suppose the n × n matrix A has eigenvalues λ1, . . . , λn ordered by
|λ1| > |λ2| > |λ3| ≥ ··· ≥ |λn|,
with linearly independent eigenvectors v(1), v(2), . . . , v(n).
a. Show that if the Power method is applied with an initial vector x(0) given by
x(0) = β2v(2) + β3v(3) +· · ·+ βnv(n),
then the sequence {μ(m)} described in Algorithm 9.1 will converge to λ2.
b. Show that for any vector x = ∑ni=1 βiv(i), the vector x(0) = (A − λ1I) x satisfies the property given in part (a).
c. Obtain an approximation to λ2 for the matrices in Exercise 1.
d. Show that this method can be continued to find λ3 using x(0) = (A − λ2I) (A − λ1I)x.