91Ó°ÊÓ

Let \(A\) be a nilpotent matrix (that is, \(A^{m}=O\) for some \(m>1\) ). Show that \(\lambda=0\) is the only eigenvalue of \(A\).

The matrices in either are not diagonalizable or do not have a dominant eigenvalue (or both). Apply the power method anyway with the given initial vector \(\mathbf{x}_{0}\) performing eight iterations in each case. Compute the exact eigenvalues and eigenvectors and explain what is happening. $$A=\left[\begin{array}{ll} 4 & 1 \\ 0 & 4 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

Let \(A\) bean idempotent matrix (that is, \(A^{2}=A\) ). Show that \(\lambda=0\) and \(\lambda=1\) are the only possible eigenvalues of \(A\)

The matrices in either are not diagonalizable or do not have a dominant eigenvalue (or both). Apply the power method anyway with the given initial vector \(\mathbf{x}_{0}\) performing eight iterations in each case. Compute the exact eigenvalues and eigenvectors and explain what is happening. $$A=\left[\begin{array}{rr} 3 & 1 \\ -1 & 1 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

If \(\mathbf{v}\) is an eigenvector of \(A\) with corresponding eigenvalue \(\lambda\) and \(c\) is a scalar, show that \(\mathbf{v}\) is an eigenvector of \(A-c I\) with corresponding eigenvalue \(\lambda-c\)

The matrices in either are not diagonalizable or do not have a dominant eigenvalue (or both). Apply the power method anyway with the given initial vector \(\mathbf{x}_{0}\) performing eight iterations in each case. Compute the exact eigenvalues and eigenvectors and explain what is happening. $$A=\left[\begin{array}{lll} 4 & 0 & 1 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]$$

Find all (real) values of k for which \(A\) is diagonalizable. $$A=\left[\begin{array}{ll} 1 & 1 \\ 0 & k \end{array}\right]$$

The matrices in either are not diagonalizable or do not have a dominant eigenvalue (or both). Apply the power method anyway with the given initial vector \(\mathbf{x}_{0}\) performing eight iterations in each case. Compute the exact eigenvalues and eigenvectors and explain what is happening. $$A=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]$$

The power method does not converge to the dominant eigenvalue and eigenvector. Verify this, using the given initial vector \(\mathbf{x}_{0} .\) Compute the exact eigenvalues and eigenvectors and explain what is happening. $$A=\left[\begin{array}{ll} -1 & 2 \\ -1 & 1 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

Find all (real) values of k for which \(A\) is diagonalizable. $$A=\left[\begin{array}{ll} 1 & k \\ 0 & 1 \end{array}\right]$$