Matrix norm

In mathematics, a norm is a function from a vector space to non-negative real numbers that satisfies certain axioms. A matrix norm is a norm defined on a vector space of matrices. As for every norm, a matrix norm defines a distance, the distance between two matrices being the norm of their difference. The specificity of matrix norms is that they may often be defined either directly as for every norm, or by mean of their properties as linear operators.

There are many specific matrix norms. Most arise from the following three perspectives, though different perspectives may sometimes yield the same norm.

• Consider the matrix as a linear operator; then a matrix norm may describe how much the operator can stretch vectors. Such matrix norms induced by vector norms are called operator norms.
• Consider the matrix as a rectangular array of numbers; then a matrix norm may be defined as a function of the entries of the matrix. Such matrix norms are sometimes called "entry-wise" norms.
• The singular value decomposition is useful in analyzing matrices. A vector norm of the singular values of a matrix may be taken as a matrix norm. Such norms are called Schatten norms.

Matrix norms are often denoted by double vertical bars with optional subscripts (e.g., $\|A\|$ or $\|A\|_{2}$). However, the meaning of the subscript may vary, since matrix norms in different perspectives relate to $\ell ^{p}$-norms in different ways.

Matrix norms in different perspectives

Operator norms	"Entry-wise" norms	Schatten norms	Also known as
induced from $\ell ^{1}$-norm	maximum column sum
		$\ell ^{1}$-norm (sum of singular values)	nuclear norm
	square root of the sum of squares	$\ell ^{2}$-norm of singular values	Frobenius norm
induced from $\ell ^{2}$-norm		$\ell ^{\infty }$-norm (the largest singular value)	spectral norm
induced from $\ell ^{\infty }$-norm	maximum row sum