Matrix Norms

Let $A_{n\times n}$ . A norm of $A$ is denoted by $||A||$ .

Definitions

Suppose $A=(a_{ij})_{m\times n}$ for all the definitions below.

1-Norm

Maximum of the absolute column sums.

{\lVert A \rVert}_1 = \max \bigg\{ \sum_{i=1}^{m} {\lvert a_{ij} \rvert} \;;\; j \in [1,n] \bigg\}

2-Norm

Square root of the sum of all elements squared. Aka. Euclidean norm, or Frobenius norm. Defined for non-square matrices as well.

\Big({\lVert A \rVert}_2\Big)^2= \Big({\lVert A \rVert}_E\Big)^2= \sum_{i=1}^{m} \sum_{j=1}^{n} {(a_{ij})^2}

Infinity Norm

Maximum of the absolute row sums.

{\lVert A \rVert}_\infty = \max \bigg\{ \sum_{j=1}^{n} {\lvert a_{ij} \rvert} \;;\; i \in [1,n] \bigg\}

Vector Norm

Norm defined for column vectors.

Induced Norm

Also known as operator norm, subordinate norm. Suppose $A=(a_{ij})_{m\times n}$ . The induced norm is defined for $A$ with respect to a given norm, $\lVert \rVert$ .

{\lVert A \rVert}_\text{ind} = \max_{\lVert X \rVert = 1}\; \lVert AX \rVert

Properties of Norms

Works for all types of norms.

Suppose $A,B$ are $m\times n$ ordered.

$\lVert A \rVert \ge 0$
$\lVert A \rVert = 0 \iff A=0$
$\lVert kA \rVert=\lvert k\rvert\times \lVert A \rVert$
$\lVert A + B\rVert \le \lVert A \rVert + \lVert B \rVert$ (triangle inequality)
$\lVert AB \rVert \le \lVert A\rVert\times \lVert B\rVert$