CAS EXPERIMENT. Orthogonal Matrices. spectra. Apply it to the matrix in Prob. 9 (call it

(Verification) Verify the statements in Example 1.

Verify the statements in Examples 3 and 4.

Are the eigenvalues of A + B of the form \(\lambda_{j}+\mu_{j}\), where \(\lambda_{j}\) and \(\mu_{j}\) are the eigenvalues of A and B, respectively? Text Transcription: lambda_j + mu_j lambda_j mu_j

(Orthogonality) Prove that eigenvectors of a symmetric matrix corresponding to different eigenvalues are orthogonal. Give an example.

(Skew-symmetric matrix) Show that the inverse of a skew-symmetric matrix is skew-symmetric.

Do there exist nonsingular skew-symmetric n X n matrices with odd n?

(Orthogonal matrix) Do there exist skew-symmetric orthogonal 3 X 3 matrices?

(Symmetric matrix) Do there exist nondiagonal symmetric 3 X 3 matrices that are orthogonal?

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{cc}0.96 -0.28 \\ 0.28 0.96\end{array}\right]\) Text Transcription: [0.96 -0.28 \\ 0.28 0.96]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{rr}a b \\ -b a\end{array}\right]\) Text Transcription: [a b \\ -b a]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{rr}3 1 \\ -1 1\end{array}\right]\) Text Transcription: [3 1 \\ -1 1]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{ccc}\cos \theta -\sin \theta 0 \\ \sin \theta \cos \theta 0 \\ 0 0 1\end{array}\right]\) Text Transcription: [cos theta - sin theta 0 \\ sin theta cos theta 0 \\ 0 0 1]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{rrr}14 4 -2 \\ 4 14 2 \\ -2 2 17\end{array}\right]\) Text Transcription: [14 4 -2 \\ 4 14 2 \\ -2 2 17]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{rrr}0 -6 -12 \\ 6 0 -12 \\ 12 12 0\end{array}\right]\) Text Transcription: [0 -6 -12 \\ 6 0 -12 \\ 12 12 0]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{rrr}0 0 1 \\ 0 1 0 \\ -1 0 0\end{array}\right]\) Text Transcription: [0 0 1 \\ 0 1 0 \\ -1 0 0]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{rrr}\frac{4}{9} \frac{8}{9} \frac{1}{9} \\ -\frac{7}{9} \frac{4}{9} -\frac{4}{9} \\ -\frac{4}{9} \frac{1}{9} \frac{8}{9}\end{array}\right]\) Text Transcription: [ 4 / 9 8 / 9 1 / 9 \\ - 7 / 9 4 / 9 - 4 / 9 \\ - 4 / 9 1 / 9 8 / 9]

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find their spectrum (thereby illustrating Theorems 1 and 5). (Show the details of your work.) \(\left[\begin{array}{lll}a b b \\ b a b \\ b b a\end{array}\right]\) Text Transcription: [a b b \\ b a b \\ b b a]

(Rotation in space) Give a geometric interpretation of the transformation y = Ax with A as in Prob. 12 and x and y referred to a Cartesian coordinate system.

WRITING PROJECT. Section Summary. Summarize the main concepts and facts in this section, with illustrative examples of your own.

Orthogonal Matrices. (a) Products. Inverse. Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations? (b) Rotation. Show that (6) is an orthogonal transformation. Verify that it satisfies Theorem 3 . Find the inverse transformation. (c) Powers. Write a program for computing powers \(\mathbf{A}^{m}(m=1,2, \cdots)\) of a 2 X 2 matrix A and their spectra. Apply it to the matrix in Prob. 9 (call it A). To what rotation does A correspond? Do the eigenvalues of \(\mathbf{A}^{m}\) have a limit as \(m \rightarrow x\) ? (d) Compute the eigenvalues of \((0.9 \mathrm{~A})^{m}\), where A is the matrix in Prob. 9. Plot them as points. What is their limit? Along what kind of curve do these points approach the limit? (e) Find A such that y = Ax is a counterclockwise rotation through \(30^{\circ}\) in the plane. Text Transcription: A^{m}(m = 1, 2, cdots) A^m m rightarrow x (0.9 A)^m 30^circ