91影视

Find singular value decompositions for the matrices listed in Exercises 7 through 14. Work with paper and pencil. In each case, draw a sketch analogous to Figure 4 in the text, showing the effect of the transformation on the unit circle, in three steps.

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

Consider a quadratic form$\mathbf{q}\left(x\right)\mathbf{=}\mathbf{xxAx}$on${\mathbf{R}}^{\mathbf{n}}$and a fixed vector$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$in${\mathbf{R}}^{\mathbf{n}}$. Is the transformation

linear? If so, what is its matrix?

The function $\mathit{q}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}{\stackrel{\mathbf{鈫�}}{\mathbf{x}}}^{\mathbf{T}}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{2}\\ \mathbf{2}& \mathbf{4}\end{array}\mathbf{\right]}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$is a quadratic form.

If Ais an invertible symmetric matrix, what is the relationship between the definiteness of A and ${\mathbf{A}}^{\mathbf{-}\mathbf{1}}$?

The singular values of any triangular matrix are the absolute values of its diagonal entries.

Show that a quadratic form$\mathit{q}\left(\stackrel{鈬赌}{x}\right)\mathbf{=}\stackrel{\mathbf{鈬赌}}{\mathbf{X}}\mathbf{xA}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$of two variables is indefinite if (and only if) detA<0. Here, is a symmetric 2x2 matrix

LetLfrom ${\mathit{R}}^{\mathbf{3}}$to ${\mathit{R}}^{\mathbf{3}}$be the reflection about the line spanned by

$\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$

a. Find an orthonormal eigenbasis for L .

b. Find the matrix B of L with respect to eigenbasis.

c. Find the matrix A of L with respect to the standard basis of${\mathit{R}}^{\mathbf{3}}$ .

If the singular values of a $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix$\mathit{A}$are 3 and 4, then there must exist a unit vector$\stackrel{\mathbf{鈫�}}{\mathbf{u}}$in${\mathit{R}}^{\mathbf{2}}$such that$||A\stackrel{鈫�}{u}||\mathbf{=}\mathbf{4}$.

Find singular value decompositions for the matrices listed in Exercises 7 through 14. Work with paper and pencil. In each case, draw a sketch analogous to Figure 4 in the text, showing the effect of the transformation on the unit circle, in three steps.

$\left[\begin{array}{cc}6& 3\\ -1& 2\end{array}\right]$

Consider a symmetric$\mathbf{3}\mathbf{脳}\mathbf{3}$ matrixAwith${\mathit{A}}^{\mathbf{2}}\mathbf{=}{\mathit{l}}_{\mathbf{3}}$ . Is the linear transformation role="math" localid="1660729963116" $\mathit{T}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$necessarily the reflection about a subspace of${\mathbf{鈩�}}^{\mathbf{3}}$ ?