# Definition:Hilbert-Schmidt Norm

## Definition

Let $A = \paren {a_{i j} }_{m \times n} \in \R^{m \times n}$ be an $m \times n$ matrix.

Let $\norm {\, \cdot \,}_{HS} : \R^{m \times n} \to \R$ be a mapping such that:

$\ds \norm A_{HS} = \sqrt {\sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^m a_{ij}^2 }$

Then $\norm {\, \cdot \,}_{HS}$ is called the Hilbert-Schmidt norm.

## Source of Name

This entry was named for David Hilbert and Erhard Schmidt.