91影视

Consider a rotation$\mathbf{T}\mathbf{}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{3}}$in. (That is, A is an orthogonal 3x3matrix with determinant 1.) Show that T has a non-zero fixed point [i.e., a vector$\mathbf{T}\mathbf{}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{3}}$with$\mathbf{T}\mathbf{}\left(\stackrel{鈫�}{v}\right)\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$]. This result is known as Euler鈥檚 theorem, after the great Swiss mathematician Leonhard Euler (1707鈥�1783). Hint: Consider the characteristic polynomialrole="math" localid="1659595800447" ${\mathbf{f}}_{\mathbf{A}}\mathbf{\left(}\mathbf{位}\mathbf{\right)}$. Pay attention to the intercepts with both axes. Use Theorem 7.1.4.

Consider an $\mathbf{n}\mathbf{脳}\mathbf{n}$matrix such that the sum of the entries in each row is . Show that the vector

$\stackrel{\mathbf{鈫�}}{\mathbf{e}}\mathbf{=}\left[\begin{array}{c}1\\ 1\\ .\\ .\\ 1\end{array}\right]$

In ${\mathbf{R}}^{\mathbf{n}}$is an eigenvector of A. What is the corresponding eigenvalue?

Consider a subspace V of${\mathbf{鈩�}}^{\mathbf{n}}$with dim(V) = m.

a. Suppose the n 脳 n matrix A represents the orthogonal projection onto V. What can you say about the eigenvalues of A and their algebraic and geometric multiplicities?

b. Suppose the n 脳 n matrix B represents the reflection about V. What can you say about the eigenvalues of B and their algebraic and geometric multiplicities?

Find a basis of the linear space Vof all $\mathbf{2}\mathbf{脳}\mathbf{2}$matrices Afor whichrole="math" localid="1659530325801" $\left[\begin{array}{c}0\\ 1\end{array}\right]$is an eigenvector, and thus determine the dimension of V.

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.

$\mathbf{T}\left(f\right)\mathbf{=}\mathbf{f}\mathbf{\text{'}}\mathbf{-}\mathbf{f}\mathbf{}\mathbf{from}\mathbf{}{\mathbf{C}}^{\mathbf{鈭�}}\mathbf{}\mathbf{to}\mathbf{}\mathbf{C}\mathbf{鈭�}$

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 through 20,find all (real) eigenvalues. Then find a basis of each eigenspaces ,and diagonalize A, if you can. Do not use technology.

$\left[\begin{array}{cc}6& 3\\ 2& 7\end{array}\right]$

Is $\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$an eigenvector of$\mathbf{A}\mathbf{+}\mathbf{2}{\mathbf{I}}_{\mathbf{n}}$? If so, what is the eigenvalue?

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

$\left[\begin{array}{cc}5& -4\\ 2& 1\end{array}\right]$

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

$\mathbf{\text{3.}}\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{0}{\mathbf{.8}}^{}& \mathbf{0}\mathbf{.7}\\ \mathbf{鈭�}\mathbf{0}\mathbf{.7}& \mathbf{0}{\mathbf{.8}}^{}\end{array}\mathbf{\right]}\mathbf{,}$

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.

3.$\mathit{A}\mathbf{=}\left[\begin{array}{cc}1& 2\\ 4& 3\end{array}\right]$