91Ó°ÊÓ

Use Householder's method to place the following matrices in tridiagonal form. a. \(\left[\begin{array}{rrr}12 & 10 & 4 \\ 10 & 8 & -5 \\ 4 & -5 & 3\end{array}\right]\) b. \(\left[\begin{array}{rrr}2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2\end{array}\right]\) c. \(\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]\) d. \(\left[\begin{array}{rrr}4.75 & 2.25 & -0.25 \\ 2.25 & 4.75 & 1.25 \\\ -0.25 & 1.25 & 4.75\end{array}\right]\)

1\. Apply two iterations of the QR method without shifting to the following matrices. a. \(\left[\begin{array}{rrr}2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]\) b. \(\left[\begin{array}{lll}3 & 1 & 0 \\ 1 & 4 & 2 \\ 0 & 2 & 1\end{array}\right]\) c. \(\left[\begin{array}{rrr}4 & -1 & 0 \\ -1 & 3 & -1 \\ 0 & -1 & 2\end{array}\right]\) d. \(\left[\begin{array}{rrrr}1 & 1 & 0 & 0 \\ 1 & 2 & -1 & 0 \\ 0 & -1 & 3 & 1 \\ 0 & 0 & 1 & 4\end{array}\right]\) e. \(\left[\begin{array}{rrrr}-2 & 1 & 0 & 0 \\ 1 & -3 & -1 & 0 \\ 0 & -1 & 1 & 1 \\ 0 & 0 & 1 & 3\end{array}\right]\) f. \(\left[\begin{array}{llll}0.5 & 0.25 & 0 & 0 \\ 0.25 & 0.8 & 0.4 & 0 \\ 0 & 0.4 & 0.6 & 0.1 \\ 0 & 0 & 0.1 & 1\end{array}\right]\)

Find the eigenvalues and associated eigenvectors of the following \(3 \times 3\) matrices. Is there a set of linearly independent eigenvectors? a. \(\quad A=\left[\begin{array}{rrr}2 & -3 & 6 \\ 0 & 3 & -4 \\ 0 & 2 & -3\end{array}\right]\) b. \(\quad A=\left[\begin{array}{lll}2 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 2\end{array}\right]\) c. \(A=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]\) d. \(\quad A=\left[\begin{array}{rrr}2 & 1 & -1 \\ 0 & 2 & 1 \\ 0 & 0 & 3\end{array}\right]\)

Determine the singular values of the following matrices. a. \(\quad A=\left[\begin{array}{ll}2 & 1 \\ 1 & 0\end{array}\right]\) b. \(\quad A=\left[\begin{array}{ll}2 & 1 \\ 1 & 0 \\ 0 & 1\end{array}\right]\) c. \(A=\left[\begin{array}{rr}2 & 1 \\ -1 & 1 \\ 1 & 1 \\ 2 & -1\end{array}\right]\) d. \(\quad A=\left[\begin{array}{rrr}1 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1 \\\ 1 & 1 & -1\end{array}\right]\)

Apply two iterations of the QR method without shifting to the following matrices. a. \(\left[\begin{array}{rrr}2 & -1 & 0 \\ -1 & -1 & -2 \\ 0 & -2 & 3\end{array}\right]\) b. \(\left[\begin{array}{lll}3 & 1 & 0 \\ 1 & 4 & 2 \\ 0 & 2 & 3\end{array}\right]\) c. \(\left[\begin{array}{lllll}4 & 2 & 0 & 0 & 0 \\ 2 & 4 & 2 & 0 & 0 \\ 0 & 2 & 4 & 2 & 0 \\ 0 & 0 & 2 & 4 & 2 \\ 0 & 0 & 0 & 2 & 4\end{array}\right]\) d. \(\left[\begin{array}{rrccc}5 & -1 & 0 & 0 & 0 \\ -1 & 4.5 & 0.2 & 0 & 0 \\\ 0 & 0.2 & 1 & -0.4 & 0 \\ 0 & 0 & -0.4 & 3 & 1 \\ 0 & 0 & 0 & 1 & 3\end{array}\right]\)

Determine the singular values of the following matrices. a. \(A=\left[\begin{array}{rr}-1 & 1 \\ 1 & 1\end{array}\right]\) b. \(\quad A=\left[\begin{array}{lll}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right]\) c. \(A=\left[\begin{array}{rr}1 & -1 \\ 1 & 1 \\ 0 & 1 \\ 1 & 0 \\ -1 & 1\end{array}\right]\) d. \(\quad A=\left[\begin{array}{lll}0 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \\\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]\)

Show that the following pairs of matrices are not similar. a. \(\quad A=\left[\begin{array}{ll}1 & 1 \\ 0 & 3\end{array}\right]\) and \(\quad B=\left[\begin{array}{ll}2 & 2 \\ 1 & 2\end{array}\right]\) b. \(\quad A=\left[\begin{array}{rr}1 & 1 \\ 2 & -2\end{array}\right]\) and \(\quad B=\left[\begin{array}{rr}-1 & 2 \\ 1 & 2\end{array}\right]\) c. \(A=\left[\begin{array}{rrr}1 & 1 & -1 \\ -1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right]\) and \(\quad B=\left[\begin{array}{rrr}2 & -2 & 0 \\ -2 & 0 & 2 \\ 2 & 2 & -2\end{array}\right]\) d. \(\quad A=\left[\begin{array}{rrr}1 & 1 & -1 \\ 2 & -2 & 2 \\ -3 & 3 & 3\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 3 & 2 \\\ 0 & 1 & 0\end{array}\right]\)

Use Householder's method to place the following matrices in tridiagonal form. a. \(\left[\begin{array}{rrrr}4 & -1 & -1 & 0 \\ -1 & 4 & 0 & -1 \\ -1 & 0 & 4 & -1 \\ 0 & -1 & -1 & 4\end{array}\right]\) c. \(\left[\begin{array}{lllll}8 & 0.25 & 0.5 & 2 & -1 \\ 0.25 & -4 & 0 & 1 & 2 \\ 0.5 & 0 & 5 & 0.75 & -1 \\ 2 & 1 & 0.75 & 5 & -0.5 \\ -1 & 2 & -1 & -0.5 & 6\end{array}\right]\) d. \(\left[\begin{array}{rrrrr}2 & -1 & -1 & 0 & 0 \\ -1 & 3 & 0 & -2 & 0 \\\ -1 & 0 & 4 & 2 & 1 \\ 0 & -2 & 2 & 8 & 3 \\ 0 & 0 & 1 & 3 & 9\end{array}\right]\)

Find the eigenvalues and associated eigenvectors of the following \(3 \times 3\) matrices. Is there a set of linearly independent eigenvectors? a. \(\quad A=\left[\begin{array}{rrr}1 & 0 & 0 \\ -1 & 0 & 1 \\ -1 & -1 & 2\end{array}\right]\) b. \(\quad A=\left[\begin{array}{rrr}2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2\end{array}\right]\) c. \(\quad A=\left[\begin{array}{lll}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right]\) d. \(\quad A=\left[\begin{array}{lll}2 & 1 & 1 \\ 0 & 3 & 1 \\ 0 & 0 & 2\end{array}\right]\)

Use the Gersgorin Circle Theorem to determine bounds for the eigenvalues, and the spectral radius of the following matrices. a. \(\left[\begin{array}{rrr}1 & 0 & 0 \\ -1 & 0 & 1 \\ -1 & -1 & 2\end{array}\right]\) b. \(\left[\begin{array}{rrr}4 & -1 & 0 \\ -1 & 4 & -1 \\ -1 & -1 & 4\end{array}\right]\) c. \(\left[\begin{array}{lll}3 & 2 & 1 \\ 2 & 3 & 0 \\ 1 & 0 & 3\end{array}\right]\) d. \(\left[\begin{array}{rrr}4.75 & 2.25 & -0.25 \\ 2.25 & 4.75 & 1.25 \\\ -0.25 & 1.25 & 4.75\end{array}\right]\)