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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...

Definition10.2\(p\)-norms.

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.

Subsection10.1Exercises

Exercise10.4

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 \(\Cm\text{?}\)

Exercise10.5

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?