91Ó°ÊÓ

For the matrices A in Exercises 1 through 12, find closed formulas for ${\mathit{A}}^{\mathbf{t}}$, where t is an arbitrary positive integer. Follow the strategy outlined in Theorem 7.4.2 and illustrated in Example 2. In Exercises 9 though 12, feel free to use technology.

2.$\mathit{A}\mathbf{=}\left[\begin{array}{cc}2& 0\\ 1& 3\end{array}\right]$

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through20,find all (real)eigenvalues.Then find a basis of each eigenspaces,and diagonalize A, if you can. Do not use technology.

$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

For the matrices A in Exercises 1 through 10 , determine whether the zero state is a stable equilibriunt of the dynamical system$\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{(}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left(}\mathit{i}\mathbf{\right)}$

$\mathbf{\text{2.}}\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{−}\mathbf{1}{\mathbf{.1}}^{}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}{\mathbf{.9}}^{}\end{array}\mathbf{\right]}\mathbf{,}$

Is $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$an eigenvector of${\mathbf{A}}^{\mathbf{-}\mathbf{1}}$? If so, what is the eigenvalue?

consider the dynamical system

$\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\left[\begin{array}{cc}1.1& 0\\ 0& \mathrm{λ}\end{array}\right]\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.

Sketch a phase portrait of this system for the given values of$\mathit{λ}$:

$\mathit{λ}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}$

(a). If 2i is an eigenvalue of a real 2 × 2 matrix A, find${\mathit{A}}^{\mathbf{2}}$.

(b). Give an example of a real 2 × 2 matrix A such that all the entries of A are nonzero and 2i is an eigenvalue of A. Compute${\mathit{A}}^{\mathbf{2}}$and check that your answer agrees with part (a).

In all parts of this problem, consider an $\mathbf{n}\mathbf{×}\mathbf{n}$matrix A such that all entries are positive and the sum of the entries in each row is 1(meaning that${\mathbf{A}}^{\mathbf{T}}$ is a positive transition matrix).

A. Consider an eigenvector$\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{ofc}$with positive components. Show that the associative eigenvalue is less than or equal to 1. Hint: consider the largest entry${\mathbf{v}}_{\mathbf{i}}\mathbf{of}\stackrel{\mathbf{→}}{\mathbf{v}}$. What can you say about the${\mathbf{i}}^{\mathbf{th}}$ entry of $\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{v}}$?

B. Now we drop the requirement that the components of the eigenvector $\stackrel{\mathbf{→}}{\mathbf{v}}$be positive. Show that the associative eigenvalue is less than or equal to1in absolute value.

C. Show that $\mathbf{λ}\mathbf{=}\mathbf{-}\mathbf{1}$fails to be an eigenvalue of A, and show that the eigenvector with eigenvalue${}^{\mathbf{1}}$are the vector of the form

$\left[\begin{array}{c}c\\ c\\ M\\ c\end{array}\right]$where is nonzero.

Consider an upper triangular $\mathbf{n}\mathbf{×}\mathbf{n}$matrix Awith${\mathbf{a}}_{\mathbf{ij}}\mathbf{≠}\mathbf{0}$for$\mathbf{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{…}\mathbf{,}\mathbf{m}$and${\mathbf{a}}_{\mathbf{ij}}\mathbf{=}\mathbf{0}$for$\mathbf{i}\mathbf{=}\mathbf{m}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{…}\mathbf{n}$. Find the algebraic multiplicity of the eigenvalueof. Without using Theorem 7.3.6, what can you say about the geometric multiplicity?

Consider the dynamical system

$\stackrel{\mathbf{→}}{\mathbf{X}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\left[\begin{array}{cc}1.1& 0\\ 0& â\dots „\end{array}\right]\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.

Sketch a phase portrait of this system for the given values of $\mathit{λ}$:

$\mathit{λ}\mathbf{=}\mathbf{1}$

Consider a positive transition matrix A. Explain why 1 is an eigenvalue of A. What you can say about the other eigenvalues? Is

$\stackrel{\mathbf{→}}{\mathbf{e}}\mathbf{=}\left[\begin{array}{c}1\\ 1\\ .\\ .\\ 1\end{array}\right]$Necessarily an eigenvector?.

Hint: Consider Ex 22,29,30_.