91Ó°ÊÓ

Diagonalize the$\mathbf{13}\mathbf{×}\mathbf{13}$matrix

$\left[\begin{array}{cccccc}0& 0& 0& …& 0& 1\\ 0& 0& 0& …& 0& 1\\ 0& 0& 0& …& 0& 1\\ â‹\circledR & â‹\circledR & â‹\circledR & â‹\pm & â‹\circledR & â‹\circledR \\ 0& 0& 0& …& 0& 1\\ 1& 1& 1& ⋯& 1& 1\end{array}\right]$

(All ones in the last row and the last column, and zeros elsewhere.)

If A is an $\mathbf{n}\mathbf{×}\mathbf{n}$matrix, what is the product of its singular values ${\mathbf{σ}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{σ}}_{\mathbf{n}}$? State the product in terms of the determinant of A. For a $\mathbf{2}\mathbf{×}\mathbf{2}$matrix A, explain this result in terms of the image of the unit circle.

Show that an SVD$\mathit{A}\mathbf{=}\mathit{U}\mathit{Î\pounds }{\mathit{V}}^{\mathbf{T}}$

can be written as$\mathit{A}\mathbf{=}{\mathit{σ}}_{\mathbf{1}}{\stackrel{\mathbf{→}}{\mathbf{u}}}_{\mathbf{1}}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}^{\mathbf{T}}\mathbf{+}\mathbf{…}\mathbf{+}{\mathit{σ}}_{\mathbf{r}}{\stackrel{\mathbf{→}}{\mathbf{u}}}_{\mathbf{r}}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{r}}^{\mathbf{T}}\mathbf{.}$

For the matrix $\mathit{A}\mathbf{=}\left[\begin{array}{cc}8& -2\\ -2& 5\end{array}\right]\mathbf{,}$write$\mathit{A}\mathbf{=}\mathit{B}{\mathit{B}}^{\mathbf{T}}$as discussed in Exercise 28. See Example 1.

Consider a symmetric matrixA. If the vector$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$ is in the image of Aand $\stackrel{\mathbf{â‡\P Ä}}{\mathit{w}}$ is in the kernel of A, is$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$ necessarily orthogonal to$\stackrel{\mathbf{â‡\P Ä}}{w}$? Justify your answer.

The equation $\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{5}\mathit{x}\mathit{y}\mathbf{+}\mathbf{3}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$ defines an ellipse.

For each of the quadratic forms q listed in Exercises 1 through 3, find the matrix of q.

2.$\mathbf{q}\left(x_1,x_2\right)\mathbf{=}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}$

If \(q\left( {\vec x} \right)\) is a positive definite quadratic form, then so is \(kq\left( {\vec x} \right)\), for any scalar \(k\).

Find a decomposition$\mathit{A}\mathbf{=}{\mathit{σ}}_{\mathbf{1}}{\stackrel{\mathbf{→}}{\mathbf{u}}}_{\mathbf{1}}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}^{\mathbf{T}}\mathbf{+}{\mathit{σ}}_{\mathbf{2}}{\stackrel{\mathbf{→}}{\mathbf{u}}}_{\mathbf{2}}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}^{\mathbf{T}}$

$\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathbf{}\mathit{A}\mathbf{}\mathbf{=}\mathbf{}\left[\begin{array}{cc}6& 2\\ -7& 6\end{array}\right]$See Exercise 29 and Example 2.

Consider an orthogonal matrix Rwhose first column is$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$. Form the symmetric matrix $\mathbf{A}\mathbf{=}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}{\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}}^{\mathbf{T}}$. Find an orthogonal matrix Sand a diagonal matrix Dsuch that ${\mathbf{S}}^{\mathbf{-}\mathbf{1}}\mathbf{AS}\mathbf{=}\mathbf{D}$ . Describe Sin terms ofR.