91影视

Consider the matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right]$

Find an orthonormal eigenbasis forA.

If A is a symmetric n x n matrix, what is the relationship between the eigenvalues of A and the singular values of A?

Consider the matrix

$\left[\begin{array}{ccccc}0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\end{array}\right]$

Find an orthogonal $\mathbf{5}\mathbf{脳}\mathbf{5}$ matrix S such that ${\mathit{S}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{S}$is diagonal.

Consider a quadratic form

$\mathbf{q}\left(\stackrel{鈬赌}{x}\right)\mathbf{=}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{.}\mathbf{A}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$

where A is a symmetricnxnmatrix. Let$\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$ be a unit eigenvector of A, with associated eigenvalue $\mathit{位}$. Find $\mathit{q}\left(\stackrel{鈬赌}{v}\right)$.

Let A be a$\mathbf{2}\mathbf{脳}\mathbf{2}$ matrix and $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$a unit vector in${\mathbf{鈩�}}^{\mathbf{2}}$. Show that${\mathit{蟽}}_{\mathbf{2}}\mathbf{猢�}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{猢�}{\mathit{蟽}}_{\mathbf{1}}\mathbf{,}$

where${\mathit{蟽}}_{\mathbf{1}}\mathbf{,}{\mathit{蟽}}_{\mathbf{2}}$ are the singular values of A. Illustrate this inequality with a sketch, and justify it algebraically.

If Ais a symmetric matrix, then there must exist an orthogonal matrix Ssuch that SAS^Tis diagonal.

Let${\mathbf{J}}_{\mathbf{n}}$be the nxnmatrix with all ones on the "other diagonal" and zeros elsewhere. (In Exercises 24 and 25, we studied${\mathbf{J}}_{\mathbf{4}}$and ${\mathbf{J}}_5$, respectively.) Find the eigenvalues of${\mathbf{J}}_{\mathbf{n}}$, with their multiplicities.

Let Abe anmatrix and $\stackrel{\mathbf{鈫�}}{\mathbf{v}}$a vector in ${\mathit{R}}^{\mathbf{m}}\mathbf{.}$Show that

${\mathit{蟽}}_{\mathbf{m}}||\stackrel{鈫�}{v}||\mathbf{鈮�}||A\stackrel{鈫�}{v}||\mathbf{鈮�}{\mathit{蟽}}_{\mathbf{1}}||\stackrel{鈫�}{v}||$

where ${\mathit{蟽}}_{\mathbf{1}}\mathit{a}\mathit{n}\mathit{d}{\mathit{蟽}}_{\mathbf{m}}$are the largest and the smallest singular values of A, respectively. Compare this with Exercise 25.

Let $位$ be a real eigenvalue of an n x n matrix A. Show that

${\mathit{蟽}}_{\mathbf{n}}\mathbf{猢�}\mathbf{\left|}\mathit{位}\mathbf{\right|}\mathbf{猢�}{\mathit{蟽}}_{\mathbf{1}}\mathbf{,}$

where${\mathit{蟽}}_{\mathbf{1}}\mathit{a}\mathit{n}\mathit{d}{\mathit{蟽}}_{\mathbf{n}}$are the largest and the smallest singular values of A, respectively.

Diagonalize the $\mathit{n}\mathbf{脳}\mathit{n}$matrix

$\left[\begin{array}{cccccc}1& 0& 0& 鈰�& 0& 1\\ 0& 1& 0& 鈰�& 1& 0\\ 0& 0& 1& 鈰�& 0& 0\\ 鈰�& 鈰�& 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 1& 0& 鈰�& 1& 0\\ 1& 0& 0& 鈰�& 0& 1\end{array}\right]$.

(All ones along both diagonals, and zeros elsewhere.)