# Matrix Norm

## 1 Definition #

Consider a number field $$K$$ which is either real or complex. The matrix norm is a function $$\| \cdot \| : K^{m \times n} \to \RR$$ that satisfies the following properties:

For all scalars $$\alpha \in K$$ and for all matrices $$A,b \in K^{m \times n}$$,

• $$\|A\|\geq 0$$
• $$\|A\| = 0 \Longleftrightarrow A = 0_{m,n}$$
• $$\|\alpha A\|= |\alpha| \|A\|$$
• $$\|A+B\|\leq \|A\|+\|B\|$$

Additionally, in the case of square matrices, some (but not all) matrix norms satisfy the following sub-multiplicative condition.

• $$\|AB\|\leq \|A\|\|B\|$$

A matrix norm that satisfies this additional property is called a sub-multiplicative norm

## 2 Operator Norm #

Suppose a vector norm $$\| \cdot \|$$ on $$K^m$$ and $$K^n$$ is given, then we define the corresponding induced norm or operator norm on the space $$K^{m\times n}$$ as follows:

\begin{align} \|A\| &=\sup \left\{ \|Ax\|: x\in K^n, \|x\|=1 \right\}\\ &=\sup \left\{ \|Ax\|: x\in K^n, \|x\|\leq 1 \right\}\\ &=\sup \left\{ \frac{\|Ax\|}{\|x\|}: x\in K^n, x\neq 0 \right\} \end{align}

The last equality is usually reformed and used as an inequality:

$\|Ax\| \leq \|A\|\|x\|$

Any induced operator norm is a sub-multiplicative matrix norm. This follows from:

$\|ABx\|\leq \|A\|\|Bx\|\leq \|A\|\|B\|\|x\|$

and

$\max_{\|x\|=1} \|ABx\| = \|AB\|$

## 3 Frobenius Norm #

Frobenius norm treats an $$m \times n$$ matrix as a vector of size $$m \cdot n$$:

$\|A\|_F = \sqrt{\langle A,A\rangle _{F}}$

where $$\langle A,A\rangle_{F}$$ is the Frobenius inner product, defined as

$\langle A,A\rangle_{F} = \sum_{i,j} \overline{A_{ij}}B_{ij} = \tr \left( \overline{A^T}B \right) = \tr \left( A^{\dagger}B \right)$