91Ó°ÊÓ

Consider a quadratic form qon with symmetric matrix A, with rank A = r.Suppose that Ahas ppositive eigenvalues, if eigenvalues are counted with their multiplicities. Show that there exists an orthogonal basis ${\stackrel{\mathbf{→}}{\mathbf{w}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}{\stackrel{\mathbf{→}}{\mathbf{w}}}_{\mathbf{n}}\mathbf{}\mathbf{of}\mathbf{}{\mathbf{R}}^{\mathbf{n}}$such that $\mathbf{q}\mathbf{(}{\mathbf{c}}_{\mathbf{1}}{\stackrel{\mathbf{→}}{\mathbf{w}}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{c}}_{\mathbf{n}}{\stackrel{\mathbf{→}}{\mathbf{w}}}_{\mathbf{n}}\mathbf{)}\mathbf{=}{\mathbf{c}}_{\mathbf{p}}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{c}}_{\mathbf{p}}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{c}}_{\mathbf{p}}^{\mathbf{2}}\mathbf{-}{\mathbf{c}}_{\mathbf{p}\mathbf{+}\mathbf{1}}^{\mathbf{2}}\mathbf{-}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{-}{\mathbf{c}}_{\mathbf{r}}^{\mathbf{2}}\mathbf{.}$.Hint: Modify the approach outlined in and 65.

If q is a quadratic form on ${\mathbf{R}}^{\mathbf{n}}$with symmetric matrix A, and if is a linear transformation from ${\mathbf{R}}^{\mathbf{m}}\mathbf{}\mathbf{to}\mathbf{}{\mathbf{R}}^{\mathbf{n}}$show that the composite function $\mathbf{p}\left(\stackrel{→}{x}\right)\mathbf{=}\mathbf{q}\left(L\left(\stackrel{→}{x}\right)\right)$is a quadratic form on role="math" localid="1659689309678" ${\mathbf{R}}^{\mathbf{m}}$ Express the symmetric matrix of p in terms of R and A.

If A is a positive definite $\mathit{n}\mathbf{×}\mathit{n}$matrix, and R is any real $\mathit{n}\mathbf{×}\mathit{n}$matrix, what can you say about the definiteness of the matrix ${\mathbf{R}}^{\mathbf{T}}\mathbf{AR}$? For which matrices R is ${\mathbf{R}}^{\mathbf{T}}\mathbf{AR}$positive definite?

Determine the definiteness of the quadratic forms in Exercises 4 through 7.

6.$\mathbf{q}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{)}\mathbf{=}\mathbf{2}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{6}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}$

Find the singular values of $\mathit{A}\mathbf{=}\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$. Find a unit vector$\stackrel{\mathbf{→}}{\mathbf{v}}$such that$||A\stackrel{→}{v_1}||\mathbf{=}{\mathbf{σ}}_{\mathbf{1}}$. Sketch the image of the unit circle.

The singular value of the $\mathbf{2}\mathbf{×}\mathbf{1}$ matrix $\left[\begin{array}{c}3\\ 4\end{array}\right]$is 5.

If Ais an indefinite$\mathit{n}\mathbf{×}\mathit{n}$ matrix, andR is a real$\mathit{n}\mathbf{×}\mathit{m}\mathbf{}\mathit{r}\mathit{a}\mathit{n}\mathit{k}\mathbf{}\mathit{n}$ what can you say about the definiteness of the matrix${\mathit{R}}^{\mathbf{T}}\mathit{A}\mathit{R}$?

If is an indefinite $\mathbf{n}\mathbf{×}\mathbf{m}$matrix, and R is any real $\mathbf{n}\mathbf{×}\mathbf{m}$matrix, what can you say about the definiteness of the matrix role="math" localid="1659684209026" ${\mathbf{R}}^{\mathbf{T}}\mathbf{AR}$ ?

Determine the definiteness of the quadratic forms in Exercises 4 through 7.

7.$\mathbf{q}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}\mathbf{)}\mathbf{=}\mathbf{3}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}$

The function $\mathit{q}\left(x_1,x_2\right)\mathbf{=}\mathbf{3}{\mathit{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathbf{5}{\mathit{x}}_{\mathbf{2}}$is a quadratic form.