The Frobenius norm $\parallel A{\parallel }_F=({\sum }_{i,j}\mathrm{\mid }{a}_{ij}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ on real or complex matrices

Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$, let $m,n\in \mathbb{N}$ and let $A=(a_{ij})\in M_{m\times n}(\mathbb{F})$. The Frobenius norm of $A$ is

$\parallel A{\parallel }_F:=({\sum }_{i<m}{\sum }_{j<n}\mathrm{\mid }{a}_{ij}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2},$

where $\mathrm{\mid }\cdot \mathrm{\mid }$ is the absolute value on $\mathbb{R}$ (Absolute value in an ordered field) for $\mathbb{F}=\mathbb{R}$ and the modulus on $\mathbb{C}$ (Real and imaginary parts, complex conjugation, and modulus) for $\mathbb{F}=\mathbb{C}$, and the outer power is the nonnegative real square root.

The displayed quantity is a norm. Regard $A$ as the element of the coordinate space ${\mathbb{F}}^{mn}$ whose $j+in$-th entry is $a_{ij}$. The standard coordinate inner product of The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products satisfies $⟨A,A⟩={\sum }_{i,j}{a}_{ij}\overline{a_{ij}}$, which equals ${\sum }_{i,j}\mathrm{\mid }{a}_{ij}{\mathrm{\mid }}^2$ because $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive). Thus $\parallel A{\parallel }_F$ is exactly the inner-product norm of The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product evaluated at $A$, and the three norm axioms hold for $\parallel \cdot {\parallel }_F$ because they hold for every inner-product norm. For real matrices the same sum is ${\sum }_{i,j}{a}_{ij}^2$, since $\mathrm{\mid }x{\mathrm{\mid }}^2=x^2$ for real $x$ by the two cases of Absolute value in an ordered field.

The Frobenius norm is the Euclidean norm of the full entry list, not the largest factor by which the matrix stretches a single vector. It can coincide with an induced $2$-norm in one-row or one-column cases, but in general it is a different matrix norm from the induced operator norms used on this page. Its relation to the spectral norm, its unitary invariance and its singular-value formula are Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison.

• The name records the entry-sum definition. Different sources write $\parallel A{\parallel }_F$, $\parallel A{\parallel }_{\mathrm{F}}$, or $\parallel A{\parallel }_{2,\mathrm{H}\mathrm{S}}$; this library uses $\parallel \cdot {\parallel }_F$ exclusively and does not fold it into the induced $p$-norm notation.

• Zero-sized shapes are included. At $m=0$ or $n=0$ the double sum is the empty sum $0$, so the unique empty matrix has Frobenius norm $0$; this is the value of the inner-product norm of the zero space.