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Magazine Problem 5
Find the eigenvalues and eigenvectors of the following matrices. (Use the given \(\lambda\) or factars.) $$\left[\begin{array}{ll} 5 & -2 \\ 9 & -6 \end{array}\right]$$
Problem 5
Find the matrix \(\mathbf{A}\) in the indicated linear transformation \(y=A x,\) Explain the geometric significance of the eigenvalues and eigenvectors of \(\mathbf{A}\). Show the details. Dilatation (uniform stretching) in \(R^{2}\) by a factor 5
Problem 5
Find an cigenbasis (a basis of eigenvectors) and diagonalize. (Show the details.) $$\left[\begin{array}{rr}3 & 2 \\\\-5 & -4\end{array}\right]$$
Problem 6
Find the matrix \(\mathbf{A}\) in the indicated linear transformation \(y=A x,\) Explain the geometric significance of the eigenvalues and eigenvectors of \(\mathbf{A}\). Show the details. Counterclockwise rotation through the angle \(\pi f 2\) about the origin in \(R^{2}\)
Problem 6
Do there exist non-singular skew-symmetric \(n \times n\) matrices with odd \(n ?\)
Problem 6
Find an cigenbasis (a basis of eigenvectors) and diagonalize. (Show the details.) $$\left[\begin{array}{rr}2 & 7 \\\6 & -9\end{array}\right]$$
Problem 7
Find an cigenbasis (a basis of eigenvectors) and diagonalize. (Show the details.) $$\left[\begin{array}{lll}1 & 0 & 1 \\\0 & 3 & 2 \\\0 & 0 & 2\end{array}\right]$$
Problem 7
Do there exist skew-symmetric orthogonal \(3 \times 3\) matrices?
Problem 7
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. $$\left[\begin{array}{ll} 3 & 5 \\ 5 & 3 \end{array}\right]$$
Problem 7
Find the eigenvalues and eigenvectors of the following matrices. (Use the given \(\lambda\) or factars.) $$\left[\begin{array}{cc} 0.8 & -0.6 \\ 0.6 & 0.8 \end{array}\right]$$
