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Linear Algebra in Action

Harry Dym

Chapter 10

Singular values and related inequalities - all with Video Answers

Educators

Chapter Questions

02:25

Problem 1

Show that if $A \in \mathbb{C}^{p \times q}$, then
(10.2) $\mathcal{R}_A=\mathcal{R}_{A A^H} \quad$ and $\operatorname{rank} A^H A=\operatorname{rank} A=\operatorname{rank} A A^H$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 2

Show that if $A \in \mathbb{C}^{p \times q}$ and $1 \leq r<q$, then $\operatorname{rank} A^H A=r \Longleftrightarrow s_r>0$ and $s_{r+1}=0$.

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Problem 3

Show that if $A \in \mathbb{C}^{p \times q}$ and $\operatorname{rank} A=r$, then the nonzero singular values of $A$ coincide with the nonzero singular values of $A^H$; i.e., $s_j(A)=s_j\left(A^H\right)$ for $j=1, \ldots, r$.

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Problem 4

Show that if $A \in \mathbb{C}^{p \times q}$ is expressed in the form (10.9), then $\mathcal{R}_A=\operatorname{span}\left\{\mathbf{v}_1, \ldots, \mathbf{v}_{\mathbf{r}}\right\}$ and $\mathcal{N}_A=\operatorname{span}\left\{\mathbf{u}_{r+1}, \ldots, \mathbf{u}_q\right\}$.

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Problem 5

Show that if $A \in \mathbb{C}^{p \times q}$ is expressed in the form (10.9), then $\mathcal{R}_{A^H}=\operatorname{span}\left\{\mathbf{u}_1, \ldots, \mathbf{u}_r\right\}$ and $\mathcal{N}_{A^H}=\operatorname{span}\left\{\mathbf{v}_{r+1}, \ldots, \mathbf{v}_p\right\}$.

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Problem 6

Let $A \in \mathbb{C}^{p \times q}$ be expressed in the form (10.9) and let (10.10) $A^{\dagger}=U_1 D^{-1} V_1^H$.

Show that $A A^{\dagger} A=A^{\dagger}, A A^{\dagger} A=A,\left(A^{\dagger} A\right)^H=A^{\dagger} A$ and $\left(A A^{\dagger}\right)^H=A A^{\dagger}$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 7

Show that there exists a choice of $B \in \mathbb{C}^{p \times q}$ with rank $B \leq$ $k$ that attains equality in (10.12).

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Problem 8

Let $A \in \mathbb{C}^{p \times q}$. Show that
(10.13) $\|A\|=\max \left\{|\langle A \mathbf{x}, \mathbf{y}\rangle|: \mathbf{x} \in \mathbb{C}^q, \mathbf{y} \in \mathbb{C}^p\right.$, and $\left.\|\mathbf{x}\|=\|\mathbf{y}\|=1\right\}$.

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Problem 9

Let $A \in \mathbb{C}^{p \times q}$. Show that
$$
\|A\|=\left\|A^H\right\|=\left\|A^H A\right\|^{1 / 2}=\left\|A A^H\right\|^{1 / 2} \text {. }
$$

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02:57

Problem 10

Let $A \in \mathbb{C}^{p \times q}$ and suppose that $s_1(A) \leq 1$. Show that the matrix $I_p-A B$ is invertible for every choice of $B \in \mathbb{C}^{q \times p}$ with $s_1(B) \leq 1$ if and only if $s_1(A)<1$.

Lucía Guerrero
Lucía Guerrero
Numerade Educator
03:12

Problem 11

Let $A \in \mathbb{C}^{n \times n}$ and let $\lambda_1, \ldots, \lambda_{2 n}$ denote the eigenvalues of the matrix
$$
B=\left[\begin{array}{cc}
O & A \\
A^H & O
\end{array}\right]
$$
repeated according to their mutiplicity and indexed so that $\lambda_1 \geq \cdots \geq \lambda_{2 n}$. Express these eigenvalues in terms of the singular values of $A$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 12

Show that if $A=A^H \in \mathbb{C}^{n \times n}$ and $B=B^H \in \mathbb{C}^{n \times n}$, then $\|A B\|^2=\left\|B A^2 B\right\|$.

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25:16

Problem 13

Redo Exercises 9.17 and 9.18 using singular value decompositions.

Chris Trentman
Chris Trentman
Numerade Educator
02:42

Problem 14

In the setting of Lemma 10.6, show that if $r<q$ and $A^{\dagger}$ denotes the Moore-Penrose inverse of $A$ introduced in Exercise 10.6, then the vector
$$
\mathbf{x}=A^{\dagger} \mathbf{b}=\sum_{j=1}^r \frac{\left\langle\mathbf{b}, \mathbf{v}_j\right\rangle}{s_j} \mathbf{u}_j
$$
may be characterized as the solution of (10.20) with the smallest norm.

Nick Johnson
Nick Johnson
Numerade Educator
03:10

Problem 15

In the setting of Lemma 10.6, show that if $r=q$, then $A^H A$ is invertible and the solution $\mathrm{x}$ of equation (10.20) given by formula (10.19) may be expressed as $\mathbf{x}=\left(A^H A\right)^{-1} A^H \mathbf{b}$.

Chris Trentman
Chris Trentman
Numerade Educator
00:51

Problem 16

Show that if $a_{i j}$ denote the entries of the matrix
$$
A=\left[\begin{array}{lll}
0 & 1 & 2 \\
3 & 1 & 0 \\
0 & 2 & 1
\end{array}\right], \quad \text { then } \max _i\left\{\min _j\left[a_{i j}\right]\right\} \neq \min _j\left\{\max _i\left[a_{i j}\right]\right\} .
$$

Fuzail Shakir
Fuzail Shakir
Numerade Educator

Problem 17

Show that if the eigenvalues of $A=A^H$ are ordered so that $\lambda_1 \geq \cdots \geq \lambda_n$, then formulas (10.24) and (10.25) hold.

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Problem 18

Show that if $A \in \mathbb{C}^{n \times n}$ is a Hermitian matrix with eigenvalues $\lambda_1 \leq \cdots \leq \lambda_n$ and $\mathcal{X}^{\perp}$ denotes the orthogonal complement of $\mathcal{X}$ in $\mathbb{C}^n$, then
$$
\lambda_{n-j+1}=\min _{\mathcal{X} \in \mathcal{S},} \max \left\{\frac{\langle A \mathbf{x}, \mathbf{x}\rangle}{\langle\mathbf{x}, \mathbf{x}\rangle}: \mathbf{x} \in \mathcal{X}^{\perp} \quad \text { and } \quad \mathbf{x} \neq \mathbf{0}\right\} \text { for } j=1, \ldots, n \text {. }
$$

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Problem 19

Show that if $A \in \mathbb{C}^{n \times n}$ is a Hermitian matrix with eigenvalues $\lambda_1 \leq \cdots \leq \lambda_n$ and $\mathcal{X}^{\perp}$ denotes the orthogonal complement of $\mathcal{X}$ in $\mathbb{C}^n$, then
$$
\lambda_j=\max _{\mathcal{X} \in \mathcal{S}_j} \min \left\{\frac{\langle A \mathbf{x}, \mathbf{x}\rangle}{\langle\mathbf{x}, \mathbf{x}\rangle}: \mathbf{x} \in \mathcal{X}^{\perp} \quad \text { and } \quad \mathbf{x} \neq \mathbf{0}\right\} \quad \text { for } \quad j=1, \ldots, n .
$$

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07:32

Problem 20

Let $A \in \mathbb{C}^{n \times n}, B \in \mathbb{C}^{n \times n}$ and let $s_j(A B)$ and $s_j(A)$ denote the singular values of the matrices $A B$ and $A$, respectively. Show that $s_j(A B) \leq\|B\| s_j(A)$.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:06

Problem 21

Let $A \in \mathbb{C}^{n \times n}$ be a Hermitian matrix with eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$. Show that $\lambda_n \leq \min a_{i i} \leq \max a_{i i} \leq \lambda_1$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 22

Verify formula (10.26).

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Problem 23

Verify the integral representation for $\varphi(x)$, assuming that $\varphi(x), \varphi^{\prime}(x)$ and $\varphi^{\prime \prime}(x)$ are nice continuous functions that tend to zero quickly enough as $x \rightarrow-\infty$ so that the integrals referred to in the following hint converge.

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02:06

Problem 24

Let $A \in \mathbb{C}^{n \times n}$, let $\beta_1, \ldots, \beta_n$ and $\gamma_1, \ldots, \gamma_n$ denote the eigenvalues of the Hermitian matrices
$$
B=\left(A+A^H\right) / 2 \text { and } C=\left(A-A^H\right) /(2 i),
$$
respectively, and let $\lambda \in \sigma(A)$. Show that
$$
\beta_1 \leq \frac{\lambda+\bar{\lambda}}{2} \leq \beta_n \quad \text { and } \quad \gamma_1 \leq \frac{\lambda-\bar{\lambda}}{2 i} \leq \gamma_n .
$$

Nick Johnson
Nick Johnson
Numerade Educator