91Ó°ÊÓ

Find an cigenbasis (a basis of eigenvectors) and diagonalize. (Show the details.) $$\left[\begin{array}{ccc}-6 & -6 & 10 \\\\-5 & -5 & 5 \\\\-9 & -9 & 13\end{array}\right]$$

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. $$\left[\begin{array}{ll} 0.4 & 0.8 \\ 0.8 & 0.4 \end{array}\right]$$

Do there exist non-diagonal symmetric \(3 \times 3\) matrices that are orthogonal?

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

Find the eigenvalues and eigenvectors of the following matrices. (Use the given \(\lambda\) or factars.) $$\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right]$$

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. $$\left[\begin{array}{ll} 2.5 & 1.5 \\ 1.5 & 6.5 \end{array}\right]$$

Find an cigenbasis (a basis of eigenvectors) and diagonalize. (Show the details.) $$\left[\begin{array}{rrr}3 & 10 & -15 \\\\-18 & 39 & 9 \\ -24 & 40 & -15\end{array}\right]$$

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. $$\left[\begin{array}{ll} 5 & 4 \\ 4 & 11 \end{array}\right]$$

Are the matrices in Probe \(5-11\) Hermitian? Skew Hermitian? Unitary? Find their eigenvalues (thereby verifying Theorem 1 ) and eigenvectors. $$\left[\begin{array}{ccc}0 & 1+i & 0 \\\1-i & 0 & 1+i \\\0 & 1-i & 0\end{array}\right]$$

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. $$\left[\begin{array}{cc} 7 & \sqrt{6} \\ \sqrt{6} & 2 \end{array}\right]$$