- A
*vector norm on matrices*, i.e, a norm on the vector space of all real or complex*m*-by-*n*matrices. - A
*sub-multiplicative vector norm*refers to a vector norm on square matrices compatible with matrix multiplication in the sense that

In the remaining article, we will follow the tradition in matrix theory. We use term "vector norm" for the first definition and "matrix norm" for the second definition.

Table of contents |

2 Operator norm or Induced norm 3 Spectral norm or Spectral radius 4 Frobenius norm |

*r*|*A*|_{1}≤ |*A*|_{2}≤*s*|*A*|_{1}

Moreover, when *m*=*n*, then for any vector norm | · |, there exists a unique positive number *k* such that *k*| · | is a (submultiplicative) matrix norm.

A matrix norm || · || is said to be *minimal* if there exists no other matrix norm | · | satisfying |A|≤||A|| for all |A|.

Spectral norm is the only minimal matrix norm which is an induced norm. The spectral norm of *A* equals to the square root of the spectral radius of *AA*_{*} or the largest singular value of *A*.

An important property for matrix norm is