Exercise 7.2.21

Answers

1.

The Corollary after 5.15 implies that

A^{m}

is again a transition matrix. So all its entries are no greater than

1

2.

Use the inequality in Exercise 7.2.20(d) and the previous argument. We compute

∥ J^{m} ∥ = ∥ P^{- 1} A^{m} P ∥ \leq n^{2} ∥ P^{- 1} ∥ ∥ A^{m} ∥ ∥ P ∥ \leq n^{2} ∥ P^{- 1} ∥ ∥ P ∥ .

Pick the fixed value $c = n^{2} ∥ P^{- 1} ∥ ∥ P ∥$ and get the result.

3.

By the previous argument, we know the norm

∥ J^{m} ∥

is bounded. If

J_{1}

is a block corresponding to the eigenvalue

1

and the size of

J_{1}

is greater than

1

, then the

12

-entry of

J_{1}^{m}

is unbounded. This is a contradiction.

4.

By Corollary 3 after Theorem 5.16, the absolute value of eigenvalues of

A

is no greater than

1

. So by Theorem 5.13, the limit

\lim_{m \to \infty} A^{m}

exists if and only if

1

is the only eigenvalue of

A

5.

Theorem 5.19 confirm that

\dim (E_{1}) = 1

. And Exercise 7.2.21(c) implies that

K_{1} = E_{1}

. So the multiplicity of the eigenvalue

1

is equal to

\dim (K_{1}) = \dim (E_{1}) = 1

by Theorem 7.4(c).

jephianlin

2011-06-27 00:00