Math Problems Vault

Problem 5.1.1

(a)(b)

Find the 1-, 2-, and $\infty$ -norms of $x = \begin{pmatrix} 2 \\ 1 \\ -4 \\ -2 \end{pmatrix}$ and $x = \begin{pmatrix} 1+i \\ 1-i \\ 1 \\ 4i \end{pmatrix}$ .

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(a)

(b)

Problem 5.1.2

(a)(b)(c)

Consider the euclidean norm with $u = \begin{pmatrix} 2 \\ 1 \\ -4 \\ -2 \end{pmatrix}$ and $v = \begin{pmatrix} 1 \\ -1 \\ 1 \\ -1 \end{pmatrix}$ .

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(a)

Determine the distance between $u$ and $v$ .

(b)

Verify that the triangle inequality holds for $u$ and $v$ .

(c)

Verify that the CBS inequality holds for $u$ and $v$ .

Problem 5.1.3

Show that $\left(\alpha_{1} + \alpha_{2} + \cdots + \alpha_{n}\right)^{2} \leq n \left(\alpha_{1}^{2} + \alpha_{2}^{2} + \cdots + \alpha_{n}^{2}\right)$ for $\alpha_{i} \in \mathbb {R}$ .

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Problem 5.1.4

(a)(b)

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(a)

Using the euclidean norm, describe the solid ball in $\mathbb {R}^{n}$ centered at the origin with unit radius.

(b)

Describe a solid ball centered at the point $c = \left(\xi_{1} \; \xi_{2} \; \cdots \; \xi_{n}\right)$ with radius $\rho$ .

Problem 5.1.5

If $x, y \in \mathbb {R}^{n}$ such that $\left\| x-y\right\|_{2} = \left\| x+y\right\|_{2}$ , what is $x^{T}y$ ?

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Problem 5.1.6

Explain why $\left\| x-y\right\| = \left\| y-x\right\|$ is true for all norms.

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Problem 5.1.7

For every vector norm on $\mathbb {C}^{n}$ , prove that $\left\| v\right\|$ depends continuously on the components of $v$ in the sense that for each $\varepsilon > 0$ , there corresponds a $\delta > 0$ such that $\left|\left\| x\right\| - \left\| y\right\| \right| < \varepsilon$ whenever $\left|x_{i} - y_{i}\right| < \delta$ for each $i$ .

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Problem 5.1.8

(a)(b)

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(a)

For $x \in \mathbb {C}^{n \times 1}$ , explain why $\left\| x\right\|_{1} \geq \left\| x\right\|_{2} \geq \left\| x\right\|_{\infty }$ .

(b)

For $x \in \mathbb {C}^{n \times 1}$ , show that $\left\| x\right\|_{i} \leq \alpha \left\| x\right\|_{j}$ , where $\alpha$ is the $(i,j)$ -entry in the following matrix. (See Exercise 5.12.3 for a similar statement regarding matrix norms.)

\begin{pmatrix} [ccc] & 1 & 2 & \infty \\ 1 & * & \sqrt{n} & n \\ 2 & * & * & \sqrt{n} \\ \infty & * & * & * \end{pmatrix}

Problem 5.1.9

For $x, y \in \mathbb {C}^{n}$ , $x \neq 0$ , explain why equality holds in the CBS inequality if and only if $y = \alpha x$ , where $\alpha = x^{*}y / x^{*}x$ .

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Remark.

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Formula (5.1.4), from the proof of the CBS inequality, states that for $\alpha = x^{*}y/x^{*}x$ ,

0 \leq \left\| \alpha x - y\right\| _{2} = \left\| y\right\| _{2}\left\| x\right\| _{2} - \frac{(x^{*}y)(y^{*}x)}{\left\| x\right\| _{2}}.

Problem 5.1.10

For nonzero vectors $x, y \in \mathbb {C}^{n}$ with the euclidean norm, prove that equality holds in the triangle inequality if and only if $y = \alpha x$ , where $\alpha$ is real and positive.

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Problem 5.1.11

Use Holder's inequality

\left|x^{*}y\right| \leq \left\| x\right\| _{p}\left\| y\right\| _{q}

(where $p>1$ , $q>1$ , $1/p+1/q=1$ ) to prove that if the components of $x \in \mathbb {R}^{n \times 1}$ sum to zero (i.e., $x^{T}e = 0$ for $e^{T} = (1,1,\ldots ,1)$ ), then

\left|x^{T}y\right| \leq \left\| x\right\| _{1} \left(\frac{y_{\max }-y_{\min }}{2}\right)

for all $y \in \mathbb {R}^{n \times 1}$ .

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Remark.

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For "zero sum" vectors $x$ , this bound is at least as sharp, and usually sharper, than Holder's inequality itself, because $(y_{\max }-y_{\min })/2 \leq \max_{i} \left|y_{i}\right| = \left\| y\right\|_{\infty }$ .

Problem 5.1.12

(a)(b)(c)

The classical form of Holder's inequality states that if $p>1$ and $q>1$ are real numbers such that $1/p+1/q=1$ , then

\sum _{i=1}^{n} \left|x_{i}y_{i}\right| \leq \left(\sum _{i=1}^{n} \left|x_{i}\right|^{p}\right)^{1/p} \left(\sum _{i=1}^{n} \left|y_{i}\right|^{q}\right)^{1/q}.

Derive this inequality by executing the following steps:

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(a)

By considering the function $f(t) = (1-\lambda ) + \lambda t - t^{\lambda }$ for $0 < \lambda < 1$ , establish the inequality

\alpha ^{\lambda }\beta ^{1-\lambda } \leq \lambda \alpha + (1-\lambda )\beta

for nonnegative real numbers $\alpha$ and $\beta$ .

(b)

Let $\hat{x} = x/\left\| x\right\|_{p}$ and $\hat{y} = y/\left\| y\right\|_{q}$ , and apply the inequality of part (a) to obtain

\sum _{i=1}^{n} \left|\hat{x}_{i}\hat{y}_{i}\right| \leq \frac{1}{p}\sum _{i=1}^{n} \left|\hat{x}_{i}\right|^{p} + \frac{1}{q}\sum _{i=1}^{n} \left|\hat{y}_{i}\right|^{q} = 1.

(c)

Deduce the classical form of Holder's inequality, and then explain why this means that $\left|x^{*}y\right| \leq \left\| x\right\|_{p}\left\| y\right\|_{q}$ .

Problem 5.1.13

The triangle inequality $\left\| x+y\right\|_{p} \leq \left\| x\right\|_{p}+\left\| y\right\|_{p}$ for a general $p$ -norm is really the classical Minkowski inequality, which states that for $p \geq 1$ ,

\left(\sum _{i=1}^{n} \left|x_{i}+y_{i}\right|^{p}\right)^{1/p} \leq \left(\sum _{i=1}^{n} \left|x_{i}\right|^{p}\right)^{1/p} + \left(\sum _{i=1}^{n} \left|y_{i}\right|^{p}\right)^{1/p}.

Derive Minkowski's inequality.

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