LMIs in Control/Matrix and LMI Properties and Tools/Quadratic Inequalities

Required data
Consider $$W\in\mathbb{S}^{n}$$, $$x, y\in\mathbb{R}^{n}$$, and $$\gamma\in\mathbb{R}_{\geqslant0}$$, where $$W>0$$.

Matrix inequality
The inequality $$(x-y)^{T}W(x-y)\leq\gamma$$ is equivalent to the matrix inequality given by

$$\begin{bmatrix} \gamma & (x-y)^{T} \\ * & W^{ -1} \end{bmatrix}\geq0$$.

Required data
Consider $$W\in\mathbb{S}^{n}$$, $$A\in\mathbb{R}^{n \times m}$$, $$x$$, $$c\in\mathbb{R}^{m}$$, $$b\in\mathbb{R}^{n}$$, and $$d\in\mathbb{R}$$, where $$W>0$$.

Matrix inequality
The quadratic inequality $$(Ax+b)^{T}W(Ax+b)-c^{T}x-d\leq0$$ with $$W>0$$ is equivalent to the matrix inequality given by

$$\begin{bmatrix} W^{ -1} & Ax+b \\ * & c^{T}x+d \end{bmatrix}\geq0$$.