In Exercises 25 and 26, mark each statement True or False.

Determine which of the matrices in Exercises 1–6 are symmetric. \(\left[\begin{array}{rr} 3 & 5 \\ 5 & -7 \end{array}\right]\)

Problem 2E Determine which of the matrices in are symmetric.

Determine which of the matrices in Exercises 1–6 are symmetric. \(\left[\begin{array}{ll}2 & 3 \\ 2 & 4\end{array}\right]\)

Problem 4E Determine which of the matrices in are symmetric.

Determine which of the matrices in Exercises 1–6 are symmetric. \(\left[\begin{array}{rrr}-6 & 2 & 0 \\ 2 & -6 & 2 \\ 0 & 2 & -6\end{array}\right]\)

Problem 39E [M] Orthogonally diagonalize the matrices in Exercises 37–40. To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and, for each eigenvalue , find an orthonormal basis for , as in Examples 2 and 3 Reference Examples 2: If possible, diagonalize the matrix Reference Examples 3:

Problem 6E Determine which of the matrices in are symmetric.

Problem 7E Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.

Problem 8E Determine which of the matrices in are orthogo¬nal. If orthogonal, find the inverse.

Problem 9E Determine which of the matrices in are orthogo¬nal. If orthogonal, find the inverse.

Problem 10E Determine which of the matrices in are orthogo¬nal. If orthogonal, find the inverse.

Problem 11E Determine which of the matrices in are orthogo¬nal. If orthogonal, find the inverse.

Problem 12E Determine which of the matrices in are orthogo¬nal. If orthogonal, find the inverse.

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix P and a diagonal matrix D. To save you ime, the eigenvalues in Exercises 17–22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5. \(\left[\begin{array}{ll}3 & 1 \\ 1 & 3\end{array}\right]\)

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17–22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5. \(\left[\begin{array}{rr}1 & -5 \\ -5 & 1\end{array}\right]\)

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17–22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5. \(\left[\begin{array}{ll} 3 & 4 \\ 4 & 9 \end{array}\right]\)

Problem 18E Orthogonally diagonalize the matrices in giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17-22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5.

Problem 17E Orthogonally diagonalize the matrices in giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17-22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5.

Problem 16E Orthogonally diagonalize the matrices in giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17-22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5.

Problem 19E Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17–22 are:

Problem 20E Orthogonally diagonalize the matrices in giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17-22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5.

Problem 21E Orthogonally diagonalize the matrices in giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17-22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5.

Problem 23E Let Verify that 5 is an eigenvalue of A and v is an eigenvector. Then orthogo¬nally diagonalize A.

Problem 22E Orthogonally diagonalize the matrices in giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17-22 are: (17) -4, 4, 7; (18) -3, -6, 9; (19) -2, 7; (20) -3, 15; (21) 1, 5, 9; (22) 3, 5.

Problem 25E In Exercises 25 and 26, mark each statement True or False. Justify each answer. a. An n × n matrix that is orthogonally diagonalizable must be symmetric. c. An n × n symmetric matrix has n distinct real eigenvalues. d. For a nonzero v in Rn, the matrix vvT is called a projection matrix.

Problem 24E Let v2= Verify that v1 and v2 are eigenvectors of A. Then 1 orthogonally diagonalize A.

Problem 26E a. There are symmetric matrices that are not orthogonally diagonalizable. b. If B = PDPT, where PT = P-l and D is a diagonal matrix, then B is a symmetric matrix. c. An orthogonal matrix is orthogonally diagonalizable. d. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

Problem 27E Show that if A is an n × n symmetric matrix, then

Problem 28E Suppose A is a symmetric n × n matrix and B is any n × m matrix. Show that BTAB, BTB, and BBT are symmetric matrices.

Suppose A is invertible and orthogonally diagonalizable. Explain why \(A^{-1}\) is also orthogonally diagonalizable.

Problem 30E Suppose A and B are both orthogonally diagonalizable and AB = BA. Explain why AB is also orthogonally diagonalizable.

Problem 32E Suppose A = PRP–1, where P is orthogonal and R is upper triangular. Show that if A is symmetric, then R is symmetric and hence is actually a diagonal matrix.

Problem 33E Construct a spectral decomposition of A from Example 2. Reference: If possible, diagonalize the matrix

Problem 31E Let A = PDP–1, where P is orthogonal and D is diagonal, and let be an eigenvalue of A of multiplicity k. Then appears k times on the diagonal of D. Explain why the dimension of the eigenspace for is k.

Problem 34E Construct a spectral decomposition of A from Example 3. Reference:

Problem 35E Let u be a unit vector in Rn, and let B = uuT . a. Given any x in Rn, compute Bx and show that Bx is the orthogonal projection of x onto u, as described in Section 6.2. b. Show that B is a symmetric matrix and B2 = B. c. Show that u is an eigenvector of B. What is the corresponding eigenvalue?

Problem 36E Let B be an n × n symmetric matrix such that B2 = B. Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any y in a. Show that z is orthogonal to . b. Let W be the column space of B. Show that y is the sum of a vector in W and a vector in . Why does this prove that By is the orthogonal projection of y onto the column space of B?