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Algebra
Matrix norms and conditioning
Frobenius norm

24. Matrix norms and conditioning

Frobenius norm

Level: intermedio

In 24. Matrix norms and conditioning

For

A=(a_{ij})\in\mathbb R^{m\times n}

\|A\|_F = \sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^2}.

Via the singular values

\|A\|_F = \sqrt{\sum_i\sigma_i^2}.

Tags

algebra
algebra formula
linear
norms
norms conditioning

Detail

Extends the Euclidean norm to the set of all matrix entries.

Explore the idea

Intuitive explanation

Trata la matriz como un vector largo formado por todas sus entradas y calcula su longitud euclidiana: suma de cuadrados y raíz.

Derivation

\|A\|_F=\sqrt{\sum_{i,j}|a_{ij}|^2}

es la norma

\ell_2

del vector

\mathrm{vec}(A)

. Con SVD,

\|A\|_F^2=\sum_i\sigma_i^2

.

Worked example

Para

A=\begin{pmatrix}1&2\\3&4\end{pmatrix}

se tiene

\|A\|_F=\sqrt{1+4+9+16}=\sqrt{30}\approx5{,}48

.

Also known as

Frobenius norm
norma euclidiana de matriz
Frobenius norm formula
matrix Frobenius norm
norma de Frobenius fórmula

Related topics

Matrix norms and distances

Last reviewed: 2026-09-14

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