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

{\displaystyle {\begin{aligned}\ X^{T}Y+Y^{T}X\leq X^{T}S^{-1}X+Y^{T}SY,\\\end{aligned}}}

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.

{\displaystyle {\begin{aligned}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{aligned}}}

which is named Young’s relation.

## Reformulation of Young’s Relation

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

{\displaystyle {\begin{aligned}\ X^{T}Y+Y^{T}X\leq {\frac {1}{2}}(X+SY)^{T}S^{-1}(X+SY),\\\end{aligned}}}

is a reformulation of Young’s relation.