Skip to main content
\(\newcommand{\Cs}{\mathbb{Cs}} \newcommand{\Cm}{\mathbb{C}^{m}} \newcommand{\Cmm}{\mathbb{C}^{m\times m}} \newcommand{\Cmn}{\mathbb{C}^{m\times n}} \newcommand{\R}{\mathbb{R}} \newcommand{\Rm}{\mathbb{R}^{m}} \newcommand{\Rn}{\mathbb{R}^{n}} \newcommand{\Rmm}{\mathbb{R}^{m\times m}} \DeclareMathOperator{\Range}{range} \DeclareMathOperator{\Rank}{rank} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\Null}{null} \newcommand{\definiteintegral}[4]{\int_{#1}^{#2}\,#3\,d#4} \newcommand{\indefiniteintegral}[2]{\int#1\,d#2} \graphicspath{ {images/} } \usepackage{amsmath} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section10Vector Norms (NM, 25 Sep)

Definition10.1Norm on \(\Cm\)

A function \(\| \cdot \|\) is called a norm on \(\Cm\) if, for all vectors \(x\) and \(y\) in \(\Cm\)

  1. \(\|x\| \geq 0\text{,}\) and \(\|x\|=0\) if and only if \(x=0\text{.}\)
  2. \(\|\lambda x\| = |\lambda| \| x\|\) for any scalar \(\lambda \in \Cs\text{.}\)
  3. \(\|x + y\| \leq \| x\| + \| y\|\) (triangle inequality).

The norm of a vector gives us some information about the “size” of the vector. But there are different ways of measuring the size: you could take the absolute value of the largest entry, you cold look at the “distance” for the origin, etc...


For any \(p \lt 1\text{,}\) define

\begin{equation*} \|x\|_p = \big( \sum_{i=1}^m|x_i|^p\big)^{1/p}. \end{equation*}

Of particular importance are the norms

  1. The 1-norm (or “taxi-cab norm”): \(\|\vec x\|_1 = \sum_{i=1}^n |x_i|\text{.}\)
  2. The \(2\)-norm (a.k.a. the Euclidean norm): \(\|x\|_2 = \bigg(\sum_{i=1}^n |x_i|^2\bigg)^{1/2}.\) Note, if \(x \in \Cm \text{,}\) then \(x^\star x = \|x\|_2^2 \text{.}\)
  3. The \(\infty\)-norm (also known as the max-norm): \(\|\vec x\|_\infty = \max_{i=1}^n |x_i|\text{.}\)

It takes a little bit of effort to show that \(\| \cdot \|_2\) satisfies the triangle inequality. First we need the Cauchy-Schwarz inequality.

This can then be applied to show that

\begin{equation*} \| x + y \|_2 \leq \| x\|_2 + \|y\|_2. \end{equation*}

It follows directly that \(\| \cdot\|_2\) is a norm.



Define the function \(\| \cdot \|_{A,1} : \Cm \to \R\) as \(\|x\|_{A,1} = \|Ax\|_1\) where \(\|\cdot \|_1\) is the usual 1-norm (a.k.a. “the taxicab norm”) on \(\Cm\text{,}\) and \(A \in \Cmm\) is non-singular. Is it true that \(\|Ax\|_1\) is a norm on \(\Cm\text{?}\)


Show that the following inequalities hold for all vectors \(x \in \Cm\text{.}\) If possible, give a nontrivial example for which equality holds.

  1. \(\|x\|_\infty \leq \| x\|_2\)
  2. \(\|x\|_2 \leq \sqrt{m}\|x\|_\infty\)
  3. \(\|x\|_2 \leq \sqrt{\|x\|_1 \|x\|_\infty}\)
  4. \(\|x\|_\infty \leq \|x\|_2 \leq \|x\|_1\)

Which, if any, of these inequalities extend to the matrix norms induced by these vector norms?