91影视

Matrix $\left[\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right]$is negative definite.

16. Matrix $\left[\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right]$ is negative definite.

Sketch the curves defined in Exercises 15 through 20. In each case, draw and label the principal axes, label the intercepts of the curve with the principal axes, and give the formula of the curve in the coordinate system defined by the principal axes.

17.$\mathbf{3}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{1}$

Consider an $\mathit{n}\mathbf{脳}\mathit{m}$ matrix A with rank$\left(A\right)\mathbf{=}\mathit{m}$and a singular value decomposition role="math" localid="1660726414811" $\mathbf{A}\mathbf{=}\mathbf{U}\underset{}{\overset{}{鈭�{\mathbf{v}}^{\mathbf{T}}}}$Show that the least-squares solution of a linear system $\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$can be written as

$\stackrel{\mathbf{鈫�}\mathbf{*}}{\mathbf{x}}\mathbf{=}\frac{\stackrel{\mathbf{鈫�}}{\mathbf{b}}\mathbf{鈥�}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}}{{\mathbf{蟽}}_{\mathbf{1}}}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\frac{\stackrel{\mathbf{鈫�}}{\mathbf{b}}\mathbf{鈥�}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}}{{\mathbf{蟽}}_{\mathbf{m}}}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{m}}$.

Use the approach of Exercise 16to find the determinant of the nxn matrix Bthat has p鈥檚 on the diagonal and q鈥檚 elsewhere:

$\mathit{B}\mathbf{=}\left[\begin{array}{cccc}p& q& 鈥�& q\\ q& p& 鈥�& q\\ 鈰�& 鈰�& 鈰�& 鈰�\\ q& q& 鈥�& p\end{array}\right]$

All skew-symmetric matrices are diagonalizable ( over $\mathit{R}$).

Sketch the curves defined in Exercises 15 through 20. In each case, draw and label the principal axes, label the intercepts of the curve with the principal axes, and give the formula of the curve in the coordinate system defined by the principal axes.

18.$\mathbf{9}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{路}\mathbf{4}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{6}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$

Consider the $\mathbf{4}\mathbf{脳}\mathbf{2}$ matrix

$\mathit{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{10}}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& 1& -1& -1\\ 1& -1& 1& -1\\ 1& -1& -1& 1\end{array}\right]\left[\begin{array}{cc}2& 0\\ 0& 1\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}3& -4\\ 4& 3\end{array}\right]$

Use the result of Exercise 17 to find the least-squares solution of the linear system

$\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$where$\stackrel{\mathbf{鈫�}}{\mathbf{b}}\mathbf{=}\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]$

Work with paper and pencil.

Consider unit vectors${\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{n}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{n}}$insuch that the angle between ${\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{i}}$and ${\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_j$is $$\begin{gathered} \mathbf{60}\mathbf{掳}\mathbf{}\mathbf{for}\mathbf{}\mathbf{all}\mathbf{}\mathbf{i}\mathbf{鈮�}\mathbf{j}\mathbf{.}\mathbf{}\mathbf{find}\mathbf{}\mathbf{the}\mathbf{}\mathbf{n}\mathbf{-}\mathbf{volume}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{n}\mathbf{-}\mathbf{parallelepiped}\mathbf{}\mathbf{spanned} \\ \mathbf{by}\mathbf{}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{n}}\mathbf{.}\mathbf{Hint}\mathbf{:}\mathbf{}\mathbf{Let}\mathbf{}\mathbf{A}\mathbf{=}\left[{\stackrel{鈬赌}{v}}_1,...,{\stackrel{鈬赌}{v}}_n\right]\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\mathbf{think}\mathbf{}\mathbf{about}\mathbf{}\mathbf{the}\mathbf{}{\mathbf{matrixA}}^{\mathbf{T}}\mathbf{A}\mathbf{}\mathbf{and}\mathbf{}\mathbf{its} \\ \mathbf{determinant}\mathbf{.}\mathbf{}\mathbf{Exercise}\mathbf{}\mathbf{17}\mathbf{}\mathbf{is}\mathbf{}\mathbf{useful}\mathbf{.} \end{gathered}$$for all.

Sketch the curves defined in Exercises 15 through 20. In each case, draw and label the principal axes, label the intercepts of the curve with the principal axes, and give the formula of the curve in the coordinate system defined by the principal axes.

19.${\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$