91Ó°ÊÓ

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

Find the eigenvalues and eigenvectors of the following matrices. (Use the given \(\lambda\) or factars.) $$\left[\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array}\right]$$

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

Find the eigenvalues and eigenvectors of the following matrices. (Use the given \(\lambda\) or factars.) $$\left[\begin{array}{rrr} 85 & -28 & -28 \\ -10 & -11 & -11 \\ -46 & -2 & -2 \end{array}\right]$$

Given \(\mathbf{A}\) in a deformation \(\mathbf{y}=\mathbf{A x},\) find the principal directions and corresponding factors of extension or contraction, Show the details. $$\left[\begin{array}{ll} 5 & 2 \\ 2 & 13 \end{array}\right]$$

Given \(\mathbf{A}\) in a deformation \(\mathbf{y}=\mathbf{A x},\) find the principal directions and corresponding factors of extension or contraction, Show the details. $$\left[\begin{array}{rr} -2 & 3 \\ 3 & -2 \end{array}\right]$$

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

Verify this for \(A\) and \(\hat{A}=P^{-1} A P\). Find eigenvectors \(y\) of A. Show that \(x=\) Py are eigenvectors of \(\hat{A}\). (Show the details of your work.) $$\mathbf{A}-\left[\begin{array}{rr}-5 & 0 \\\0 & 2\end{array}\right], \mathbf{P}-\left[\begin{array}{rr}4 & -2 \\\\-3 & 1\end{array}\right]$$

Is the given matrix (call it A) Hermitian or skew-Hermitian? Find \(\mathbf{x}^{\top} \mathbf{A} \mathbf{x},\) (Show all the details) \(a, b, c, k\) are real. $$\left[\begin{array}{cc}0 & -3 t \\\\-3 i & 0\end{array}\right], x=\left[\begin{array}{c}4+l \\\3-i\end{array}\right]$$

Is the given matrix (call it A) Hermitian or skew-Hermitian? Find \(\mathbf{x}^{\top} \mathbf{A} \mathbf{x},\) (Show all the details) \(a, b, c, k\) are real. $$\left[\begin{array}{cc}a & b+t c \\\b-i c & k\end{array}\right] \cdot \mathbf{x}-\left[\begin{array}{l}x_{1} \\\x_{2}\end{array}\right]$$