Page 564 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(a) What can you say about the dimensions of the eigenspaces of A?

(b) What can you say about the dimensions of the eigenspaces if you know that A is
diagonalizable?

(c) If is a linearly independent set of eigenvectors of A all of which correspond

to the same eigenvalue of A, what can you say about the eigenvalue?

27. (For Readers Who Have Studied Calculus) If , , …, , … is an infinite sequence of

matrices, then the sequence is said to converge to the matrix A if the entries in the ith

row and jth column of the sequence converge to the entry in the ith row and jth column of A for

all i and j. In that case we call A the limit of the sequence and write . The

algebraic properties of such limits mirror those of numerical limits. Thus, for example, if P is an

invertible matrix whose entries do not depend on k, then if and only if

.

(a) Suppose that A is an diagonalizable matrix. Under what conditions on the
eigenvalues of A will the sequence A, , …, , … converge? Explain your reasoning.

(b) What is the limit when your conditions are satisfied?

28. (For Readers Who Have Studied Calculus) If is an infinite series of

matrices, then the series is said to converge if its sequence of partial sums converges to

some limit A in the sense defined in Exercise 27. In that case we call A the sum of the series and

write .

(a) From calculus, under what conditions on x does the geometric series

converge? What is the sum?

(b) Judging on the basis of Exercise 27, under what conditions on the eigenvalues of A would

you expect the geometric matrix series to converge? Explain

your reasoning.

(c) What is the sum of the series when it converges?

Show that the Jordan block matrix has as its only eigenvalue and that the corresponding

29. eigenspace is span .