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Linear Algebra in Action

Harry Dym

Chapter 17

Matrix valued holomorphic functions - all with Video Answers

Educators

Chapter Questions

01:40

Problem 1

Verify directly that
$\lim _{\xi \rightarrow 0} \frac{(\lambda+\xi)^n-\lambda^n}{\xi}=n \lambda^{n-1}$ for every positive integer $\quad n \geq 2$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator

Problem 2

Use the rules of contour integration to calculate the integral (17.10) when $f(\lambda)=\lambda$ and (a) $\gamma(t)=t$ for $1 \leq t \leq 2$; (b) $\gamma(t)=t^2$ for $1 \leq t \leq \sqrt{2}$; (c) $\gamma(t)=e^t$ for $0 \leq t \leq \ln 2$ and (d) $\gamma(t)=1+\sin t$ for $0 \leq t \leq \pi / 2$.

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01:07

Problem 3

Use the rules of contour integration to calculate the integral (17.10) when $f(\lambda)=\lambda$ and $\Gamma$ is the rectangle directed counterclockwise with vertices $-a-i b, a-i b, a+i b,-a+i b$, where $a>0$ and $b>0$.

Raj Bala
Raj Bala
Numerade Educator
01:01

Problem 4

Repeat the preceding exercise for $f(\lambda)=\lambda^n, n$ an integer (positive, zero or negative) and the same curve $\Gamma$.

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator

Problem 5

Verify Corollary 17.9.

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Problem 6

Verify assertion (6) in Theorem 17.1.is a polynomial with $n$ distinct roots $\alpha_1, \ldots, \alpha_n$. Let $\Gamma$ be a simple closed smooth curve directed counterclockwise such that $\Gamma$ and all the points inside $\Gamma$ belong to $\Omega$. Suppose further that $\alpha_1, \ldots, \alpha_{\ell}$ are inside $\Gamma$ and $\alpha_{\ell+1}, \ldots, \alpha_n$ lie outside $\Gamma$. Then
$$
\frac{1}{2 \pi i} \int_{\Gamma} f(\lambda) d \lambda=\sum_{j=1}^{\ell} \operatorname{Res}\left(f, \alpha_j\right),
$$
where
$$
\operatorname{Res}\left(f, \alpha_j\right)=\lim _{\lambda \rightarrow \alpha_j} \frac{\left\{\left(\lambda-\alpha_j\right)^{k_j} f(\lambda)\right\}^{(k,-1)}}{\left(k_j-1\right)!},
$$
and the superscript $k_j-1$ in the formula indicates the order of differentiation.

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Problem 7

Show that
$$
\int_{-\infty}^{\infty} \frac{e^{i t x}}{x^2+1} d x=\pi e^t \quad \text { if } t<0
$$
by integrating along the curve $\Gamma_R$ shown in Figure 4.

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Problem 8

Verify the evaluation of the integral given in the preceding example by exploiting the fact that
$$
\int_{-\infty}^{\infty} \frac{1-\cos t x}{x^2} d x=\lim _{\varepsilon \downarrow 0} \int_{-\infty}^{\infty} \frac{1-\cos t x}{x^2+\varepsilon^2} d x
$$

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01:00

Problem 9

Show that if $a<b \leq c<d$, then
$$
\int_{-\infty}^{\infty} \widehat{f}_{c d}(\mu) \overline{\hat{f}_{a b}(\mu)} d \mu=0 .
$$

Raj Bala
Raj Bala
Numerade Educator

Problem 10

Show that
$$
\lim _{R \uparrow \infty} \frac{1}{2 \pi} \int_{-R}^R e^{-i \mu x} \widehat{f}_{a b}(\mu) d \mu=f_{a b}(x)
$$
for all points $x \in \mathbb{R}$ other than $a$ and $b$.
In view of the formulas in Exercises 17.9 and 17.10, it is now easy to check that
$$
\lim _{R \uparrow \infty} \frac{1}{2 \pi} \int_{-R}^R e^{-i \mu x} \widehat{f}(\mu) d \mu=f(x)
$$
and
$$
\frac{1}{2 \pi} \int_{-\infty}^{\infty}|\hat{f}(\mu)|^2 d \mu=\int_{-\infty}^{\infty}|f(x)|^2 d x
$$
for functions $f$ of the form
$$
f(x)=\sum_{j=1}^n c_j f_{a_j b_j}(x),
$$
where $a_1<b_1 \leq a_2<b_2 \leq \cdots \leq a_n<b_n$ and $c_1, \ldots, c_n$ is any set of complex numbers. The first formula (17.13) exhibits a way of recovering $f(x)$ from its Fourier transform $\hat{f}(\mu)$. Accordingly, the auxiliary transform
$$
g^{\vee}(\mu)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-i \mu x} g(\mu) d \mu
$$

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Problem 11

Show that (17.14) holds if and only if
$$
\frac{1}{2 \pi} \int_{-\infty}^{\infty} \widehat{f}(\mu) \overline{\widehat{g}(\mu)} d \mu=\int_{-\infty}^{\infty} f(x) \overline{g(x)} d x
$$
holds for every pair of piecewise constant functions $f(x)$ and $g(x)$. [HINT: This is just (8.5).]

The space $L^2$ has the pleasant feature that $f \in L^2 \Longleftrightarrow \widehat{f} \in L^2$. An even pleasanter class for Fourier analysis is the Schwartz class $\mathcal{S}$ of infinitely differentiable functions $f(x)$ on $\mathbb{R}$ such that
$$
\lim _{x \uparrow+\infty}\left|x^j f^{(k)}(x)\right|=\lim _{x \downarrow-\infty}\left|x^j f^{(k)}(x)\right|=0
$$
for every pair of nonnegative integers $j$ and $k$.

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01:23

Problem 12

Show that if $f \in \mathcal{S}$, then its Fourier transform $\widehat{f}(\lambda)$ enjoys the following properties:
(a) $(-i \lambda)^j \widehat{f}(\lambda)=\int_{-\infty}^{\infty} e^{i \lambda x} f^{(j)}(x) d x$ for $j=1,2, \ldots$.
(b) $\left(-i D_\lambda\right)^k \widehat{f}=\int_{-\infty}^{\infty} e^{i \lambda x} x^k f(x) d x$ for $k=1,2, \ldots$.
(c) $\widehat{f} \in \mathcal{S}$.

You may take it as known that if $f \in \mathcal{S}$, then the derivative
$$
D_\lambda \hat{f}=\lim _{\xi \rightarrow 0} \frac{\widehat{f}(\lambda+\xi)-\hat{f}(\lambda)}{\xi}
$$
can be brought inside the integral that defines the transform.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 13

Show that if $f(x)$ and $g(x)$ belong to the Schwartz class $\mathcal{S}$, then the convolution
$$
(f \circ g)(x)=\int_{-\infty}^{\infty} f(x-y) g(y) d y
$$
belongs to the class $\mathcal{S}$ and that
$$
\widehat{(f \circ g)}(\lambda)=\widehat{f}(\lambda) \widehat{g}(\lambda) .
$$

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02:46

Problem 14

Show that if $f(x)=e^{-x^2 / 2}$, then $\hat{f}(\mu)=e^{-\mu^2 / 2} \widehat{f}(0)$.

Linh Vu
Linh Vu
Numerade Educator

Problem 15

Show that the roots $\lambda_1, \ldots, \lambda_n$ of the polynomial $p(\lambda)=$ $a_0+a_1 \lambda+\cdots+a_n \lambda^n$ with $a_n \neq 0$ depend continuously on the coefficients $a_0, \ldots, a_n$ of the polynomial.

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Problem 16

Let $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & \beta \\ 0 & 1\end{array}\right]$, where $\beta \neq 0$. Show that $\|A-B\|=|\beta|$, but that $A$ is diagonalizable, whereas $B$ is not,

Victor Salazar
Victor Salazar
Numerade Educator
02:42

Problem 17

Let $A \in \mathbb{C}^{p \times q}$ and suppose that $\operatorname{rank} A=k$ and $k<$ $\min \{p, q\}$. Show that for every $\epsilon>0$ there exists a matrix $B \in \mathbb{C}^{p \times q}$ such that $\|A-B\|<\epsilon$ and $\operatorname{rank} B=k+1$. [HINT: Use the singular value decomposition of A.]

Nick Johnson
Nick Johnson
Numerade Educator

Problem 18

Show that $\left\{A \in \mathbb{C}^{n \times n}: A\right.$ is invertible $\}$ is a generic set.

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Problem 19

Show that if $A \in \mathbb{C}^{n \times n}$ and $\sigma(A)$ belongs to the set of points enclosed by a simple smooth counterclockwise directed closed curve $\Gamma$, then
$$
\frac{1}{2 \pi i} \int_{\Gamma} e^\lambda\left(\lambda I_n-A\right)^{-1} d \lambda=\sum_{j=0}^{\infty} \frac{A^j}{j!} .
$$
Let $A \in \mathbb{C}^{n \times n}$ and let $f(\lambda)$ be holomorphic in an open set $\Omega$ that contains $\sigma(A)$. Then, in view of formulas (17.28) and (17.30) it is reasonable to define
$$
f(A)=\frac{1}{2 \pi i} \int_{\Gamma} f(\lambda)\left(\lambda I_n-A\right)^{-1} d \lambda,
$$
where $\Gamma$ is any simple smooth counterclockwise directed closed curve in $\Omega$ that encloses $\sigma(A)$ such that every point inside $\Gamma$ also belongs to $\Omega$. This definition is independent of the choice of $\Gamma$ and is consistent with the definitions of $f(A)$ considered earlier.

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Problem 20

Show that if, in terms of the notation introduced in (17.21), $A=U_1 C_\alpha^{(p)} V_1+U_2 C_\beta^{(q)} V_2$, then
$$
f(A)=U_1 \sum_{j=0}^{p-1} \frac{f^{(j)}(\alpha)}{j!}\left(C_0^{(p)}\right)^j V_1+U_2 \sum_{j=0}^{q-1} \frac{f^{(j)}(\alpha)}{j!}\left(C_0^{(q)}\right)^j V_2
$$
for every function $f(\lambda)$ that is holomorphic in an open set that contains the points $\alpha$ and $\beta$.

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Problem 21

Show that in the setting of Exercise 17.20
$$
\operatorname{det}\left(\lambda I_n-f(A)\right)=(\lambda-f(\alpha))^p(\lambda-f(\beta))^q .
$$

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01:30

Problem 22

Show that if $A \in \mathbb{C}^{n \times n}$ and $f(\lambda)$ is holomorphic in an open set that contains $\sigma(A)$, then
$$
\begin{aligned}
\operatorname{det}\left(\lambda I_n-A\right)= & \left(\lambda-\lambda_1\right)^{\alpha_1} \cdots\left(\lambda-\lambda_k\right)^{\alpha_k} \\
& \Longrightarrow \operatorname{det}\left(\lambda I_n-f(A)\right)=\left(\lambda-f\left(\lambda_1\right)\right)^{\alpha_1} \cdots\left(\lambda-f\left(\lambda_k\right)\right)^{\alpha_k} .
\end{aligned}
$$

Sam Sohn
Sam Sohn
Numerade Educator
01:39

Problem 23

Let $A \in \mathbb{C}^{n \times n}$. Show that if $A \succ O$, then
$$
A^{1 / 2}=\frac{1}{2 \pi i} \int_{\Gamma} \sqrt{\lambda}\left(\lambda I_n-A\right)^{-1} d \lambda
$$
for any simple closed smooth curve $\Gamma$ in the open right half plane that includes the eigenvalues of $A$ in its interior.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 24

Let $A, B \in \mathbb{C}^{n \times n}$ and suppose that $A \succ B \succ O$. Show that if $0<t<1$, then
$$
A^t-B^t=\frac{1}{2 \pi i} \int_{\Gamma} \lambda^t\left\{\left(\lambda I_n-A\right)^{-1}-\left(\lambda I_n-B\right)^{-1}\right\} d \lambda,
$$
where $\Gamma$ indicates the curve in Figure 7, and then, by passing to appropriate limits, obtain the formula
$$
A^t-B^t=\frac{\sin \pi t}{\pi} \int_0^{\infty} x^t\left(x I_n+A\right)^{-1}(A-B)\left(x I_n+B\right)^{-1} d t
$$

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Problem 25

Use formula (17.37) to show that if $A, B \in \mathbb{C}^{n \times n}$, then
$$
A \succ B \succ O \Longrightarrow A^t \succ B^t \text { for } 0<t<1 \text {. }
$$

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