Frobenius Norm

• categories: Linear algebra, Norm

Definition

The Frobenius norm of a matrix $A\in {\mathbb{R}}^{m\times n}$ is defined as:

$$\Vert A{\Vert }_F=\sqrt{\sum _{i=1}^m\sum _{j=1}^n|A_{ij}{|}^2}.$$

Alternatively:

$$\Vert A{\Vert }_F=\sqrt{\text{tr}(A^{T}A)}=\sqrt{\sum _{i=1}^r{\sigma }_i^2},$$

where:

• $\text{tr}(\cdot )$ is the trace operator,
• $r=\text{rank}(A)$,
• ${\sigma }_i$ are the singular values of $A$.

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Properties

1. Non-Negativity:
   $\Vert A{\Vert }_F\ge 0$, with equality only if $A=0$.

2. Scaling:
   $\Vert \alpha A{\Vert }_F=|\alpha |\Vert A{\Vert }_F$ for $\alpha \in \mathbb{R}$.

3. Triangle Inequality:
   $\Vert A+B{\Vert }_F\le \Vert A{\Vert }_F+\Vert B{\Vert }_F$.

4. Unitary Invariance:
   For unitary $U$ and $V$, $\Vert UAV{\Vert }_F=\Vert A{\Vert }_F$.

5. Relation to Vector Norm:
   Treating $A$ as a vector by concatenating rows or columns gives:

   $$\Vert A{\Vert }_F=\Vert \text{vec}(A){\Vert }_2.$$

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Examples

Example 1: Frobenius Norm of a Small Matrix

Let:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right].$$

Compute:

$$\Vert A{\Vert }_F=\sqrt{1^2+2^2+3^2+4^2}=\sqrt{30}.$$

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Example 2: Using Singular Values

For $A\in {\mathbb{R}}^{3\times 2}$ with singular values ${\sigma }_1=5$ and ${\sigma }_2=3$:

$$\Vert A{\Vert }_F=\sqrt{{\sigma }_1^2+{\sigma }_2^2}=\sqrt{25+9}=\sqrt{34}.$$