# LMIs in Control/Matrix and LMI Properties and Tools/Frobenius Norm

## Frobenius Norm

Consider ${\displaystyle A\in \mathbb {R} ^{n\times m}}$  and ${\displaystyle \gamma \in \mathbb {R} }$ >0

The Frobenius norm of ${\displaystyle A}$  is ${\displaystyle ||A||}$ F =${\displaystyle {\sqrt {tr(A^{T}A)}}={\sqrt {tr(AA^{T})}}}$

The Frobenius norm is less than or equal to ${\displaystyle \gamma }$  if and only if any of the following equivalent conditions are satisfied.

1.There exists ${\displaystyle S\in \mathbb {R} ^{n}}$  such that

${\displaystyle {\begin{bmatrix}Z&A^{T}\\*&1\\\end{bmatrix}}\geq 0,}$
{\displaystyle {\begin{aligned}\qquad tr(Z)\leq \gamma ^{\text{2}}.\qquad \\\end{aligned}}}

2.There exists ${\displaystyle S\in \mathbb {R} ^{m}}$  such that

${\displaystyle {\begin{bmatrix}Z&A\\*&1\\\end{bmatrix}}\geq 0,}$
{\displaystyle {\begin{aligned}\qquad tr(Z)\leq \gamma ^{\text{2}}.\qquad \\\end{aligned}}}