---
tag: JT8Q204E4R
kind: concept
name: Frobenius norm
msc: [15A60]
specializes: [54T6Y6DSY4]
sources:
  - url: https://en.wikipedia.org/wiki/Matrix_norm
    locator: Frobenius norm
---

## Summary

The Frobenius norm of a matrix is the square root of the sum of the squares of the absolute values of its entries.
It is the Euclidean length of the matrix seen as a long vector, and it comes from the inner product $\ip{A}{B} = \tr(B^* A)$.
It is a submultiplicative [[matrix norm|54T6Y6DSY4]] that is easy to compute.
It equals the square root of the sum of the squares of the [[singular values|09VMDSRXWJ]].

## Explanations

### L1 human

Write all entries of a matrix in one long row and measure the length of that row as in the theorem of Pythagoras.
That length is the Frobenius norm.
For $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ it is $\sqrt{1 + 4 + 9 + 16} = \sqrt{30}$.
It is used to measure how far one matrix is from another, for example in least squares fitting and low rank approximation.