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

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

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

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${\mathit{R}}^{\mathbf{3}}$ .

Show that the diagonal elements of a positive definite matrix A are positive.

The determinant of a negative definite$\mathbf{4}\mathbf{×}\mathbf{4}$matrix must be positive.

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

All positive definite matrices are invertible.

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

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.

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.

Show that any matrix of rank rcan be written as the sum of r matrices of rank 1.