Definition:Frobenius Norm

Definition

Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let $\map \MM {m, n}$ denote the vector space of matrices of order $m \times n$ over the complex numbers $\C$.

Definition 1

The Frobenius norm on $\map \MM {m, n}$ is defined and denoted as:

$\forall \mathbf A \in \map \MM {m, n}: \norm {\mathbf A}_F := \sqrt {\map \tr {\mathbf A^\dagger \mathbf A} }$

where:

$\tr$ denotes the trace of a matrix

$\mathbf A^\dagger$ denotes the Hermitian conjugate of $\mathbf A$.

Definition 2

The Frobenius norm on $\map \MM {m, n}$ is defined and denoted as:

$\forall \mathbf A \in \map \MM {m, n}: \norm {\mathbf A}_F := \sqrt {\map \tr {\mathbf A \mathbf A^\dagger} }$

where:

$\tr$ denotes the trace of a matrix

$\mathbf A^\dagger$ denotes the Hermitian conjugate of $\mathbf A$.

Definition 3

The Frobenius norm on $\map \MM {m, n}$ is defined and denoted as:

$\forall \mathbf A \in \map \MM {m, n}: \norm {\mathbf A}_F := \ds \sqrt {\sum_{i \mathop = 1}^m \sum_{j \mathop = 1}^n \cmod {a_{i j} }^2}$

where $a_{i j}$ denotes the $\tuple {i, j}$th element of $\mathbf A$.

That is, the Frobenius norm is the square root of the sum of the squares of the moduli of all the elements of $\mathbf A$.

Also known as

The Frobenius norm is also known as the Euclidean norm, after Euclid.

Also see

• Equivalence of Definitions of Frobenius Norm

• Results about the Frobenius norm can be found here.

Source of Name

This entry was named for Ferdinand Georg Frobenius.