91Ó°ÊÓ

Let Rbe a complex upper triangular nxn matrix with $\left|r_{\mathrm{ii}}\right|\mathbf{<}\mathbf{}\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathit{i}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{n}\mathbf{}$. Show that

$\underset{\mathbf{t}\mathbf{→}\mathbf{∞}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{R}}^{\mathbf{t}}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{,}$,

meaning that the modulus of all entries of ${\mathit{R}}^{\mathbf{t}}$approaches zero. Hint: We can write $\left|R\right|\mathbf{≤}\left(l_n+U\right)$, for some positive real number $\left|\mathrm{λ}\right|\mathbf{<}\mathbf{1}$and an upper triangular U > 0 matrixwith zeros on the diagonal. Exercises 47 and 48 are helpful.

If the matrix $\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$ is positive definite, then$\mathbf{\text{'}}\mathit{a}\mathbf{\text{'}}$must be positive.

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

4.$\mathbf{q}\left(x_1,x_2\right)\mathbf{=}\mathbf{6}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{3}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}$

Find the singular values of$\mathbf{A}\mathbf{=}\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

Similar matrices must have the same singular values.

Let be a complex $\mathit{n}\mathbf{×}\mathit{n}$matrix such that$\left|\mathrm{λ}\right|\mathbf{<}\mathbf{1}$ for all eigenvalues$\mathit{λ}$ of . Show that role="math" localid="1659610526426" $\underset{\mathbf{t}\mathbf{→}\mathbf{∞}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{A}}^{\mathbf{t}}\mathbf{=}\mathbf{0}$, meaning that the modulus of all entries of${\mathit{A}}^{\mathbf{t}}$ approaches zero.

b. Prove Theorem 7.6.2.

Consider the linear transformation

$\mathbf{T}\mathbf{=}\mathbf{\left(}\mathbf{q}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{)}\mathbf{\right)}\mathbf{=}{\mathbf{x}}_{\mathbf{1}}\frac{\mathbf{∂}\mathbf{q}}{\mathbf{∂}{\mathbf{x}}_{\mathbf{2}}}\mathbf{+}{\mathbf{x}}_{\mathbf{2}}\frac{\mathbf{∂}\mathbf{q}}{\mathbf{∂}{\mathbf{x}}_{\mathbf{1}}}$

from ${\mathit{Q}}_{\mathbf{2}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathit{Q}}_{\mathbf{2}}$. Find all the eigenvalues and eigenfunctions of T . Is transformation T diagonalizable?

51. IfAis a symmetric$\mathbf{2}\mathbf{×}\mathbf{2}$ matrix with eigenvalues 1 and 2, then the angle between$\stackrel{\mathbf{→}}{\mathbf{x}}$and$\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}$ must be less than$\mathbf{π}\mathbf{/}\mathbf{6}$, for all nonzero vectors$\stackrel{\mathbf{→}}{\mathbf{x}}$in${\mathbf{R}}^{\mathbf{2}}$.

54. If Aand B are real symmetric matrices such that,${\mathbf{A}}^{\mathbf{3}}\mathbf{=}{\mathbf{B}}^{\mathbf{3}}$ thenmust be equal to B.

If A is a positive semidefinite matrix with ${\mathit{a}}_{\mathbf{11}}\mathbf{=}\mathbf{0}$, what can you say about the other entries in the first row and in the first column of A? Hint: Exercise 53 is helpful.