91Ó°ÊÓ

Find the SVD of the following matrices by hand calculation: (a) \(\left[\begin{array}{ll}3 & 0 \\ 4 & 0\end{array}\right]\) (b) \(\left[\begin{array}{rr}6 & -2 \\ 8 & \frac{3}{2}\end{array}\right]\) (c) \(\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]\) (d) \(\left[\begin{array}{rr}-4 & -12 \\ 12 & 11\end{array}\right]\) (e) \(\left[\begin{array}{rr}0 & -2 \\ -1 & 0\end{array}\right]\)

Find the characteristic polynomial and the eigenvalues and eigenvectors of the following matrices: (a) \(\left[\begin{array}{rrr}1 & 0 & 1 \\ 0 & 3 & -2 \\ 0 & 0 & 2\end{array}\right]\) (b) \(\left[\begin{array}{rrr}1 & 0 & -\frac{1}{3} \\ 0 & 1 & \frac{2}{3} \\\ -1 & 1 & 1\end{array}\right]\) (c) \(\left[\begin{array}{rrr}-\frac{1}{2} & -\frac{1}{2} & -\frac{1}{6} \\ -1 & 0 & \frac{1}{3} \\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2}\end{array}\right]\)

Prove that a square matrix and its transpose have the same characteristic polynomial, and therefore the same set of eigenvalues.

Assume that \(A\) is a \(3 \times 3\) matrix with the given eigenvalues. Decide to which eigenvalue Inverse Power Iteration with the given shift \(s\) will converge, and determine the convergence rate constant \(S\). (a) \(\\{3,1,4\\}, s=5\) (b) \(\\{3,1,-4\\}, s=4\) (c) \(\\{-1,2,4\\}, s=1\) (d) \(\\{1,9,10\\}, s=8\)

Let \(A=\left[\begin{array}{ll}1 & 2 \\ 4 & 3\end{array}\right]\). (a) Find all eigenvalues and eigenvectors of \(A\). (b) Apply three steps of Power Iteration with initial vector \(x_{0}=(1,0)\). At each step, approximate the eigenvalue by the current Rayleigh quotient. (c) Predict the result of applying Inverse Power Iteration with shift \(s=0\) (d) with shift \(s=3\).