91Ó°ÊÓ

In Exercises \(I-6,\) show that \(\mathbf{v}\) is an eigenvector of \(A\) and find the corresponding eigenvalue.. $$A=\left[\begin{array}{ll} 0 & 3 \\ 3 & 0 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

Show that \(A\) and \(B\) are not similar matrices. $$A=\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 1 & 1 & 4 \end{array}\right], B=\left[\begin{array}{lll} 1 & 0 & 3 \\ 1 & 2 & 2 \\ 1 & 0 & 3 \end{array}\right]$$

If a Leslie matrix has a unique positive eigenvalue \(\lambda_{1}\) what is the significance for the population if \(\lambda_{1}>1 ?\) \(\lambda_{1}<1 ? \lambda_{1}=1 ?\)

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. $$A=\left[\begin{array}{rr} 2 & 1 \\ -2 & 5 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

It can be shown that a nonnegative \(n \times n\) matrix is irreducible if and only if \((I+A)^{n-1}>0 .\) Use this criterion to determine whether the matrix \(A\) is irreducible. If \(A\) is reducible, find a permutation of its rows and columns that puts \(A\) into the block form \\[ \left[\begin{array}{ll} B & C \\ O & D \end{array}\right] \\] $$A=\left[\begin{array}{lllll} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \end{array}\right]$$

The absolute value of a matrix \(A=\left[a_{i j}\right]\) is defined to be the matrix \(|A|=\left[\left|a_{i j}\right|\right]\) Let \(A\) and \(B\) be \(n \times n\) matrices, \(\mathbf{x}\) a vector in \(\mathbb{R}^{n},\) and \(c\) a scalar. Prove the following matrix inequalities: (a) \(|c A|=|c||A|\) (b) \(|A+B| \leq|A|+|B|\) (c) \(|A \mathbf{x}| \leq|A||\mathbf{x}|\) (d) \(|A B| \leq|A||B|\)

Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions. $$a_{1}=128, a_{n}=a_{n-1} / 2 \text { for } n \geq 2$$