Exercises

Exercises 5.9 Exercises

1.

Suppose that $A$ is an $m\times n$ matrix with rank $n\text{,}$ and that $m\ge n\text{.}$

Prove that there exists some $\lambda \gt 0$ such that $\Vert A\bx \Vert \ge \lambda \Vert\bx \Vert$ for all $\bx\in \bbR^{n}\text{.}$

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Prove that there exists some $\delta >0$ such that any $m\times n$ matrix $B$ satisfying $\Vert B-A\Vert \lt \delta$ also has rank $n\text{.}$

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2.

Suppose that $\bff$ is differentiable at $\ba \in \bbR^{n}$ and takes values in $\bbR^{m}$ with $m\ge n\text{,}$ and that $D\bff (\ba)$ has rank $n\text{.}$

Prove that there exist some $r >0$ and $\lambda > 0$ such that $\Vert \bff (\bx) -\bff (\ba)\Vert \ge \lambda \Vert \bx -\ba \Vert $ for all $\bx\in \bbR^{n}$ such that $\Vert \bx -\ba \Vert \le r\text{.}$

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Prove that there exists some $\delta >0$ such that for all $\by \in B(\bff(\ba), \delta)\text{,}$ the open ball of radius $\delta$ centered at $\bff(\ba)\text{,}$

\begin{equation*} \min_{\bx\in \overline{B(\ba, r)}} \Vert \bff (\bx) - \by \Vert \end{equation*}

is attained at some $\bx^{*} \in B(\ba, r)\text{,}$ and that

\begin{equation*} \left( \bff (\bx^{*}) - \by\right)\cdot \left( D\bff(\bx^{*})\bh \right)=0 \quad \text{ for all $\bh \in \bbR^{n}$.} \end{equation*}

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$D\bff(\bx^{*})\bh$ is a tangent vector to the image of $\bff$ at $\bff(\bx^{*})\text{,}$ so the above statement shows that $\bff(\bx^{*})$ is closest to $\by$ among $\bff(\bx), \bx\in \overline{B(\ba, r)}$ and $\bff (\bx^{*}) - \by$ is orthogonal to all tangents to the image of $\bff$ at $\bff(\bx^{*})\text{.}$
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In the set up above, assume furthermore that $D\bff(\bx)$ is continuous in $\bx\text{,}$ then prove that $r>0$ can be adjusted so that $D\bff(\bx)$ has rank $n$ for all $\bx \in B(\ba, r)\text{.}$ Assume in addition that $m=n\text{.}$ Prove that $\bff (\bx)=\by$ in the above and $r>0$ above can be adjusted so that such an $\bx$ is unique in $B(\ba, r)\text{.}$

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3.

Verify that the function $A\in \cM^{n\times n} \mapsto A^{2}$ is a differentiable function, where $\cM^{n\times n}$ is the space of $n\times n$ matrices. Then determine its derivative and check whether it is invertible in the case of $A=\begin{bmatrix} 1 \amp 0 \\ 0 \amp 1\end{bmatrix}$ and $A=\begin{bmatrix} 1 \amp 0 \\ 0 \amp -1\end{bmatrix}$ respectively.

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Show that there exist neighborhoods $U, V$ of $\begin{bmatrix} 1 \amp 0 \\ 0 \amp 1\end{bmatrix}$ such that $S:U\mapsto V$ has an inverse which is differentiable, namely, any matrix in $V$ has a uniquely defined square root in $U\text{.}$ Can one do the same around $\begin{bmatrix} 1 \amp 0 \\ 0 \amp -1\end{bmatrix}\text{?}$

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Hint.

It may not be practical to work out the Jacobian matrix in the usual matrix form, but it suffices to obtain a linear function $L(H)\in \cM^{n\times n}$ of $H\in \cM^{n\times n}$ such that $\Vert (A+H)^{2}-A^{2}-L(H)\Vert/\Vert H\Vert \to 0$ as $\Vert H\Vert \to 0\text{.}$

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4.

Consider the $N$th Fourier series partial sum $s_{N}(f; x)$ of $f\in C[-\pi, \pi]$ as a linear map from $C[-\pi, \pi]$ to itself. Using the integral expression for $s_{N}(f; x)$ to show that its operator norm is $\frac{1}{2\pi} \int_{-\pi}^{\pi} |D_{N}(t)|\,dt\text{.}$

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5.

Prove a lower bound of $\frac{\int_{x_{1}}^{x_{2}}f(x)g(x)\, dx}{\sqrt{\left( \int_{x_{1}}^{x_{2}}f(x)^{2}\,dx\right) \left( \int_{x_{1}}^{x_{2}}g(x)^{2}\,dx\right) }}$ when $f(x), g(x)$ are subject to positive upper and lower bounds. This problem is from #93 in Part II, Chapter 2 of Polya and Szeg’s classic “Problems and Theorems in Analysis I”

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Let $a, A, b, B$ be positive numbers such that $a\lt A, b\lt B\text{.}$ If the two functions $f(x) $ and $g(x)$ are integrable over the interval $[x_{1}, x_{2}]\text{,}$ and $a\le f(x)\le A, b\le g(x)\le B$ on the interval. Then

\begin{equation*} \frac{\int_{x_{1}}^{x_{2}}f(x)g(x)\, dx}{\sqrt{\left( \int_{x_{1}}^{x_{2}}f(x)^{2}\,dx\right) \left( \int_{x_{1}}^{x_{2}}g(x)^{2}\,dx\right) }}\ge \frac{2}{\sqrt{\frac{AB}{ab}+\sqrt{\frac{ab}{AB}}}}. \end{equation*}

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