Frobenius norm

Definition

For a matrix with real entries

Suppose $m,n$ are positive integers and $A$ is a $m\times n$ matrix. The Frobenius norm of $A$, denoted $\|A\|_{F}$, can be defined in the following equivalent ways:

1. It is the square root of the sum of squares of all the entries of $A$, i.e., it is the sum ${\sqrt {\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}^{2}}}$.
2. It is the square root of the trace of the $m\times m$ matrix $AA^{T}$, where $A^{T}$ is the matrix transpose of $A$.
3. It is the square root of the trace of the $n\times n$ matrix $A^{T}A$, where $A^{T}$ is the matrix transpose of $A$.
4. It is the square root of the sum of squares of the $\min\{m,n\}$ many singular values of $A$.

The Frobenius norm is invariant under orthogonal transformations (and in particular, under rotations) and is an easy-to-compute invariant.

For a matrix with complex entries

Suppose $m,n$ are positive integers and $A$ is a $m\times n$ matrix. The Frobenius norm of $A$, denoted $\|A\|_{F}$, can be defined in the following equivalent ways:

1. It is the square root of the sum of squares of the moduli of the entries of $A$, i.e., it is the sum $\sum _{i=1}^{m}\sum _{j=1}^{n}|a_{ij}|^{2}$.
2. It is the square root of the trace of the matrix $AA^{*}$ where $A^{*}$ is the matrix conjugate transpose of $A$.
3. It is the square root of the trace of the matrix $A^{*}A$ where $A^{*}$ is the matrix conjugate transpose of $A$.
4. It is the square root of the sum of squares of the $\min\{m,n\}$ many singular values of $A$.