---
tag: 54T6Y6DSY4
kind: concept
name: Matrix norm
msc: [15A60, 65F35]
specializes: [0V2WP9B1MF]
sources:
  - url: https://en.wikipedia.org/wiki/Matrix_norm
same_as:
  wikidata: Q939253
  wikipedia: Matrix norm
---

## Summary

A matrix norm is a [[norm|0V2WP9B1MF]] on a space of matrices, often required to be submultiplicative: $\norm{AB} \le \norm{A} \norm{B}$.
It measures the size of a matrix and so the size of errors and perturbations in matrix computations.
The main examples are the [[operator norms|06EKW1AKXF]] induced by vector norms and the [[Frobenius norm|JT8Q204E4R]].
Every submultiplicative matrix norm bounds the [[spectral radius|9V7YQA85PC]] from above.

## Explanations

### L1 human

A matrix norm gives one nonnegative number that says how big a matrix is.
One way is to ask how much the matrix can stretch a vector of length $1$.
Another way is to treat the matrix as a long list of numbers and take its usual length.
For the diagonal matrix with entries $3$ and $4$, the largest stretch is $4$, and the length of the list $(3, 0, 0, 4)$ is $5$.