Chapter 22, Convexity Video Solutions, Linear Algebra in Action

17:58

Problem 1

Verify that the two sets indicated just above are both convex.

Oswaldo Jiménez

Numerade Educator

Problem 2

Show that $Q=\left\{A \in \mathbb{C}^{n \times n}: A \succeq O\right\}$ is a convex set.

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Show that
$$
Q=\left\{(A, B, C) \in \mathbb{C}^{n \times n} \times \mathbb{C}^{n \times n} \times \mathbb{C}^{n \times n}:\|A\| \leq 1,\|B\| \leq 1 \text { and }\|C\| \leq 1\right\}
$$
is a convex set.

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Problem 4

Show that the four-sided figure in Figure 1 is not convex

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Problem 5

Complete the proof of Lemma 22.3.

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01:59

Problem 6

Let $f(x)=x^r$ on the set $Q=(0, \infty)$. Show that $f(x)$ is convex on $Q$ if and only if $r \geq 1$ or $r \leq 0$ and that $-f(x)$ is convex on $Q$ if and only if $0 \leq r \leq 1$.

Minh Le

Numerade Educator

Problem 7

Let $B$ be a closed nonempty convex subset of $\mathbb{R}^n$, let $\mathbf{a}_0 \in$ $\mathbb{R}^n$ and let $h(\mathbf{x})=\left\langle\mathbf{a}_0-\mathbf{b}_0, \mathbf{x}\right\rangle$, where $\mathbf{b}_0$ is the unique vector in $B$ that is closest to $\mathbf{a}_0$. Show that if $\mathbf{a}_0 \notin B$, then there exists a number $\delta>0$ such that $h\left(\mathbf{a}_0\right) \geq \delta+h(\mathbf{b})$ for every vector $\mathbf{b} \in B$.

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04:44

Problem 9

Let $Q$ be a nonempty subset of $\mathbb{R}^n$. Show that every nonzero vector $\mathbf{x} \in \operatorname{conv} Q$ can be expressed as a linear combination $\sum_{j=1}^{\ell} \alpha_j \mathbf{x}_j$ of $\ell$ linearly independent vectors in $Q$ with positive coefficients.

Gideon Idumah

Numerade Educator

Problem 10

Let $Q$ be a nonempty subset of $\mathbb{R}^n$. Show that every nonzero vector $\mathbf{x} \in K_Q$ can be expressed as a linear combination $\sum_{j=1}^{\ell} \alpha_j \mathbf{x}_j$ of $\ell$ linearly independent vectors in $Q$ with positive coefficients.

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05:58

Problem 11

Let $Q_1=\left\{\mathbf{x} \in \mathbb{R}^3: x_1^2+x_2^2 \leq 1\right.$ and $\left.x_3=0\right\}, Q_2=$ $\left\{\mathbf{x} \in \mathbb{R}^3: x_1=1, x_2=0\right.$ and $\left.-1 \leq x_3 \leq 1\right\}$. Show that the set of extreme points of the set conv $\left(Q_1 \cup Q_2\right)$ is not a closed subset of $\mathbb{R}^3$.

Chris Trentman

Numerade Educator

Problem 12

Show that the function
$$
\mathbf{f}(\mathbf{x})=\left[\begin{array}{c}
\left(x_1+x_2\right) / 2 \\
\sqrt{x_1 x_2}
\end{array}\right]
$$
maps the set $Q=\left\{\mathbf{x} \in \mathbb{R}^2: 1 \leq x_1 \leq 2\right.$ and $\left.1 \leq x_2 \leq 2\right\}$ into itself and then invoke Theorem 22.22 to establish the existence of fixed points in this set and find them.

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02:17

Problem 13

Show that the function $\mathbf{f}$ defined in Exercise 22.12 does not satisfy the constraint $\|\mathbf{f}(\mathbf{x})-\mathbf{f}(\mathbf{y})\|<\gamma\|\mathbf{x}-\mathbf{y}\|$ for all vectors $\mathbf{x}, \mathbf{y}$ in the set $Q$ that is considered there if $\gamma<1$.

Alex Roush

Numerade Educator

Problem 14

Complete the proof of Lemma 22.23 by verifying items (4) and (5).

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Problem 15

Show that in the setting of Lemma 22.23, $p_Q(\mathbf{x})<1 \Longrightarrow$ $\mathbf{x} \in Q$ and $\mathbf{x} \in Q \Longrightarrow p_Q(\mathbf{x}) \leq 1$

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Problem 16

Verify the inclusion $W(A) \subseteq \operatorname{conv}(\sigma(A))$ for normal matrices $A \in \mathbb{C}^{n \times n}$ by checking directly that every convex combination $\sum_{i=1}^n t_i \lambda_i$ of the eigenvalues $\lambda_1, \ldots, \lambda_n$ of $A$ belongs to $W(A)$.

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05:01