LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation (Completion of the Squares)

This method is used to solve quadratic equations that can't be factorized.

Matrix inequality
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY,\\ \end{align}$$ which is named Young’s relation or Young’s inequality.

Derivation
Young’s relation can be derived from a completion of the squares as follows.

\begin{align} 0 \leq (X-SY)^{T}S^{-1}(X-SY) \\ 0 \leq X^{T}S^{-1}X + Y^{T}SY - X^{T}Y - Y^{T}X\\ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY, \end{align}$$ which is named Young’s relation.

Reformulation of Young’s Relation
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq \frac12(X+SY)^{T}S^{-1}(X+SY) ,\\ \end{align}$$ is a reformulation of Young’s relation.