91Ó°ÊÓ

For each of the matrices find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology. $$\left[\begin{array}{ll} 5 & -4 \\ 2 & -1 \end{array}\right]$$

For an arbitrary positive integer \(n,\) find all complex numbers \(z\) such that \(z^{n}=1\) (in polar form). Represent your answers graphically.

Use de Moivre's formula to express \(\cos (3 \theta)\) and \(\sin (3 \theta)\) in terms of \(\cos \theta\) and \(\sin \theta.\)

Consider the complex number \(z=0.8-0.7 i .\) Represent the powers \(z^{2}, z^{3}, \ldots\) in the complex plane and explain their long-term behavior.

For the matrices \(A\) listed in Exercises 13 through 17 find an invertible matrix \(S\) such that \(S^{-1} A S=\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]\) where a and b are real numbers. $$\left[\begin{array}{ll} 1 & -2 \\ 1 & -1 \end{array}\right]$$

For each of the matrices \(A\) find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize \(A,\) if you can. Do not use technology. $$\left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

For the matrices \(A\) find \(\lim A^{t} .\) Feel free to use Theorem 7.4 .1 \(t \rightarrow \infty\). $$A=\left[\begin{array}{ll} 0.2 & 1 \\ 0.8 & 0 \end{array}\right]$$

Find all eigenvalues and eigenvectors of \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\). Is there an eigenbasis? Interpret your result geometrically.

Find all eigenvalues of the positive transition matrix \\[A=\left[\begin{array}{ll} 0.5 & 0.25 \\ 0.5 & 0.75 \end{array}\right].\\] See Definitions 2.1 .4 and 2.3 .10.

Consider an upper triangular \(n \times n\) matrix \(A\) with \(a_{i i} \neq\) 0 for \(i=1,2, \ldots, m\) and \(a_{i i}=0\) for \(i=m+1, \ldots, n\) Find the algebraic multiplicity of the eigenvalue 0 of \(A\) Without using Theorem \(7.3 .6,\) what can you say about the geometric multiplicity?