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Subsection 1.2.4 The vector p-norms

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A vector norm is a measure of the magnitude of a vector. The Euclidean norm (length) is merely the best known such measure. There are others. A simple alternative is the 1-norm.

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Definition 1.2.4.1. Vector 1-norm.

The vector 1-norm, β€–β‹…β€–1:Cmβ†’R, is defined for x∈Cm by

β€–xβ€–1=|Ο‡0|+|Ο‡1|+β‹―+|Ο‡mβˆ’1|=mβˆ’1βˆ‘i=0|Ο‡i|.
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Homework 1.2.4.1.

Prove that the vector 1-norm is a norm.

Solution

We show that the three conditions are met:

Let \(x, y \in \C^m \) and \(\alpha \in \mathbb C \) be arbitrarily chosen. Then

  • \(x \neq 0 \Rightarrow \| x \|_1 > 0 \) (\(\| \cdot \|_1 \) is positive definite):

    Notice that \(x \neq 0 \) means that at least one of its components is nonzero. Let's assume that \(\chi_j \neq 0 \text{.}\) Then

    \begin{equation*} \| x \|_1 = \vert \chi_0 \vert + \cdots + \vert \chi_{m-1} \vert \geq \vert \chi_j \vert > 0 . \end{equation*}

  • \(\| \alpha x \|_1 = \vert \alpha \vert \| x \|_1 \) (\(\| \cdot \|_1 \) is homogeneous):

    \begin{equation*} \begin{array}{l} \| \alpha x \|_1 ~~~=~~~~ \lt \mbox{ scaling~a~vector-scales-its-components; definition} \gt \\ \vert \alpha \chi_0 \vert + \cdots + \vert \alpha \chi_{m-1} \vert \\ ~~~=~~~~ \lt \mbox{ algebra} \gt \\ \vert \alpha \vert \vert \chi_0 \vert + \cdots + \vert \alpha \vert \vert \chi_{m-1} \vert \\ ~~~=~~~~ \lt \mbox{ algebra} \gt \\ \vert \alpha \vert ( \vert \chi_0 \vert + \cdots + \vert \chi_{m-1} \vert ) \\ ~~~=~~~~ \lt \mbox{ definition} \gt \\ \vert \alpha \vert \| x \|_1. \end{array} \end{equation*}

  • \(\| x + y \|_1 \leq \| x \|_1 + \| y \|_1 \) (\(\| \cdot \|_1 \) obeys the triangle inequality):

    \begin{equation*} \begin{array}{l} \| x + y \|_1 \\ ~~~=~~~~ \lt \mbox{ vector~addition;~definition~of~1-norm} \gt \\ \vert \chi_0 + \psi_0 \vert + \vert \chi_1 + \psi_1 \vert + \cdots + \vert \chi_{m-1} + \psi_{m-1} \vert \\ ~~~\leq~~~~ \lt \mbox{ algebra} \gt \\ \vert \chi_0 \vert + \vert \psi_0 \vert + \vert \chi_1 \vert + \vert \psi_1 \vert + \cdots + \vert \chi_{m-1} \vert + \vert \psi_{m-1} \vert \\ ~~~=~~~~ \lt \mbox{ commutivity} \gt \\ \vert \chi_0 \vert + \vert \chi_1 \vert + \cdots + \vert \chi_{m-1} \vert + \vert \psi_0 \vert + \vert \psi_1 \vert + \cdots + \vert \psi_{m-1} \vert \\ ~~~= ~~~~ \lt \mbox{ associativity;~definition} \gt \\ \| x \|_1 + \| y \|_1. \end{array} \end{equation*}

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The vector 1-norm is sometimes referred to as the "taxi-cab norm". It is the distance that a taxi travels, from one point on a street to another such point, along the streets of a city that has square city blocks.

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Another alternative is the infinity norm.

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Definition 1.2.4.2. Vector ∞-norm.

The vector ∞-norm, β€–β‹…β€–βˆž:Cmβ†’R, is defined for x∈Cm by

β€–xβ€–βˆž=max

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The infinity norm simply measures how large the vector is by the magnitude of its largest entry.

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Homework 1.2.4.2.

Prove that the vector \infty-norm is a norm.

Solution

We show that the three conditions are met:

Let \(x, y \in \C^m \) and \(\alpha \in \mathbb C \) be arbitrarily chosen. Then

  • \(x \neq 0 \Rightarrow \| x \|_\infty > 0 \) (\(\| \cdot \|_\infty \) is positive definite):

    Notice that \(x \neq 0 \) means that at least one of its components is nonzero. Let's assume that \(\chi_j \neq 0 \text{.}\) Then

    \begin{equation*} \| x \|_\infty = \max_{i=0}^{m-1} \vert \chi_i \vert \ge \vert \chi_j \vert > 0. \end{equation*}

  • \(\| \alpha x \|_\infty = \vert \alpha \vert \| x \|_\infty \) (\(\| \cdot \|_\infty \) is homogeneous):

    \begin{equation*} \begin{array}{lcl} \| \alpha x \|_\infty \amp =\amp \max_{i=0}^{m-1} \vert \alpha \chi_i \vert \\ \amp =\amp \max_{i=0}^{m-1} \vert \alpha \vert \vert \chi_i \vert \\ \amp =\amp \vert \alpha \vert \max_{i=0}^{m-1} \vert \chi_i \vert \\ \amp =\amp \vert \alpha \vert \| x \|_\infty. \end{array} \end{equation*}

  • \(\| x + y \|_\infty \leq \| x \|_\infty + \| y \|_\infty \) (\(\| \cdot \|_\infty \) obeys the triangle inequality):

    \begin{equation*} \begin{array}{lcl} \| x + y \|_\infty \amp =\amp \max_{i=0}^{m-1} \vert \chi_i + \psi_i \vert \\ \amp \leq\amp \max_{i=0}^{m-1} ( \vert \chi_i \vert + \vert \psi_i \vert ) \\ \amp \leq\amp \max_{i=0}^{m-1} \vert \chi_i \vert + \max_{i=0}^{m-1} \vert \psi_i \vert \\ \amp = \amp \| x \|_\infty + \| y \|_\infty. \end{array} \end{equation*}

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In this course, we will primarily use the vector 1-norm, 2-norm, and \infty-norms. For completeness, we briefly discuss their generalization: the vector p-norm.

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Definition 1.2.4.3. Vector p-norm.

Given p \geq 1 \text{,} the vector p-norm \| \cdot \|_p : \C^m \rightarrow \mathbb R is defined for x \in \C^m by

\begin{equation*} \| x \|_p = \sqrt[p]{\vert \chi_0 \vert^p + \cdots + \vert \chi_{m-1} \vert^p} = \left( \sum_{i=0}^{m-1} \vert \chi_i \vert^p \right)^{1/p}. \end{equation*}

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The proof of this result is very similar to the proof of the fact that the 2-norm is a norm. It depends on HΓΆlder's inequality, which is a generalization of the Cauchy-Schwarz inequality:

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We skip the proof of HΓΆlder's inequality and Theorem 1.2.4.4. You can easily find proofs for these results, should you be interested.

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Remark 1.2.4.6.

The vector 1-norm and 2-norm are obviously special cases of the vector p-norm. It can be easily shown that the vector \infty-norm is also related:

\begin{equation*} \lim_{p \rightarrow \infty} \| x \|_p = \| x \|_{\infty}. \end{equation*}
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Ponder This 1.2.4.3.
Consider Homework 1.2.3.3. Try to elegantly formulate this question in the most general way you can think of. How do you prove the result?
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Ponder This 1.2.4.4.

Consider the vector norm \| \cdot \|: \Cm \rightarrow \mathbb R \text{,} the matrix A \in \mathbb C^{m \times n} and the function f: \Cn \rightarrow \mathbb R defined by f( x ) = \| A x \| \text{.} For what matrices A is the function f a norm?

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