[M] Diagonalize the matrices in Exercises 33–36. Use your

Problem 3E In Exercises 3 and 4, use the factorization A = PDP–1 to compute Ak , where k represents an arbitrary positive integer.

Problem 1E In Exercises 1 and 2, let A = PDP–1 and compute A4.

Problem 4E In Exercises 3 and 4, use the factorization A = PDP–1 to compute Ak , where k represents an arbitrary positive integer.

Problem 36E [M] Diagonalize the matrices in Exercises 33–36. Use your matrix program’s eigenvalue command to find the eigenvalues, and then compute bases for the eigenspaces as in Section 5.1.

Problem 5E In Exercises 5 and 6, the matrix A is factored in the form PDP–1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.

Problem 2E In Exercises 1 and 2, let A = PDP–1 and compute A4.

Problem 6E In Exercises 5 and 6, the matrix A is factored in the form PDP–1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.

Problem 7E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 8E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 9E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 10E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 12E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 13E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 11E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 14E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 15E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 16E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 17E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 18E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 19E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 20E Diagonalize the matrices in Exercises 7–20, if possible. The real eigenvalues for Exercises 11–16 and 18 are included below the matrix.

Problem 21E In Exercises 21 and 22, A, B, P, and D are n × n matrices. Mark each statement True or False. Justify each answer. (Study Theorems 5 and 6 and the examples in this section carefully before you try these exercises.) a. A is diagonalizable if A = PDP–1 for some matrix D and some invertible matrix P. b. If Rn has a basis of eigenvectors of A, then A is diagonalizable. c. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. d. If A is diagonalizable, then A is invertible.

Problem 23E A is a 5 × 5 matrix with two eigenvalues. One eigenspace is three-dimensional, and the other eigenspace is two dimensional. Is A diagonalizable? Why?

Problem 22E In Exercises 21 and 22, A, B, P, and D are n × n matrices. Mark each statement True or False. Justify each answer. (Study Theorems 5 and 6 and the examples in this section carefully before you try these exercises.) a. A is diagonalizable if A has n eigenvectors. b. If A is diagonalizable, then A has n distinct eigenvalues. c. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. d. If A is invertible, then A is diagonalizable. Step-by-step solution

Problem 24E A is a 3 × 3 matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable? Why?

Problem 25E A is a 4 × 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other eigenspaces is two dimensional. Is it possible that A is not diagonalizable? Justify your answer.

Problem 26E A is a 7 × 7 matrix with three eigenvalues. One eigenspace is two-dimensional, and one of the other eigenspaces is three-dimensional. Is it possible that A is not diagonalizable? Justify your answer.

Problem 27E Show that if A is both diagonalizable and invertible, then so is A–1.

Problem 28E Show that if A has n linearly independent eigenvectors, then so does AT . [Hint: Use the Diagonalization Theorem.] Reference: The Diagonalization Theorem An n × n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In fact, A D PDP–1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A that correspond, respectively, to the eigenvectors in P.

Problem 29E A factorization A D PDP–1 is not unique. Demonstrate this for the matrix A in Example 2. With D1 = Use the information in Example 2 to find a matrix P1 such that A = P1 D1 P1 -1. Reference: Let Find a formula for Ak, given that A D PDP–1, where

Problem 30E With A and D as in Example 2, find an invertible P2 unequal to the P in Example 2, such that A = P2DP2-1. Reference: Let Find a formula for Ak, given that A D PDP–1, where

Problem 31E Construct a nonzero 2 × 2 matrix that is invertible but not diagonalizable.

Problem 33E [M] Diagonalize the matrices in Exercises 33–36. Use your matrix program’s eigenvalue command to find the eigenvalues, and then compute bases for the eigenspaces as in Section 5.1.

Problem 32E Construct a non-diagonal 2 × 2 matrix that is diagonalizable but not invertible.

Problem 34E [M] Diagonalize the matrices in Exercises 33–36. Use your matrix program’s eigenvalue command to find the eigenvalues, and then compute bases for the eigenspaces as in Section 5.1.

Problem 35E [M] Diagonalize the matrices in Exercises 33–36. Use your matrix program’s eigenvalue command to find the eigenvalues, and then compute bases for the eigenspaces as in Section 5.1.