# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Dualization Lemma

## Dualization Lemma

Consider ${\displaystyle P_{i}\in {\text{S}}^{n}}$ and the subspaces ${\displaystyle U,V}$, where ${\displaystyle P}$ is invertible and ${\displaystyle U+V={\text{R}}^{n}}$. The following are equivalent.

${\displaystyle X^{T}PX<0}$ for all ${\displaystyle X\in {\text{U}}}$\${\displaystyle \left\{0\right\}}$ and ${\displaystyle X^{T}PX\geq 0}$ for all ${\displaystyle X\in {\text{V}}}$.

${\displaystyle X^{T}P^{-1}X>0}$ for all ${\displaystyle X\in {\text{U}}^{\bot }}$\${\displaystyle \left\{0\right\}}$ and ${\displaystyle X^{T}P^{-1}X\leq 0}$ for all ${\displaystyle X\in {\text{V}}^{\bot }}$.

## Example

Consider the matrices ${\displaystyle Q\in {\text{S}}^{n},S\in {\text{R}}^{n\times m},R\in {\text{S}}^{m},M\in {\text{R}}^{m\times n}}$ where ${\displaystyle R\geq 0,}$ which define the quadratic matrix inequality

{\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}1&M\\\end{bmatrix}}{\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}}{\begin{bmatrix}1\\M\\\end{bmatrix}}<0.\qquad (1)\end{aligned}}}

Define {\displaystyle {\begin{aligned}P={\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}},U=R({\begin{bmatrix}0\\1\\\end{bmatrix}})\end{aligned}}} where ${\displaystyle U+V=R^{n+m}}$. Notice that ${\displaystyle (1)}$ is equivalent to ${\displaystyle X^{T}PX<0}$ for all ${\displaystyle X\in {\text{U}}}$\${\displaystyle \left\{0\right\}}$.Additionally, ${\displaystyle X^{T}PX<0\geq }$ for all ${\displaystyle X\in {\text{V}}}$ is euaivalent to

{\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}0&1\\\end{bmatrix}}{\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}}{\begin{bmatrix}0\\1\\\end{bmatrix}}=R\geq 0,\end{aligned}}}

which is satisfied based on the definition of ${\displaystyle R}$ . By the dualization lemma, ${\displaystyle (1)}$ is satisfied with ${\displaystyle R\geq 0}$ if and only if

{\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}-M^{T}&1\\\end{bmatrix}}{\begin{bmatrix}{\tilde {Q}}&{\tilde {S}}\\{\tilde {S}}^{T}&{\tilde {R}}\\\end{bmatrix}}{\begin{bmatrix}-M^{T}\\1\\\end{bmatrix}}>0,\qquad {\tilde {Q}}\leq 0,\end{aligned}}}

where {\displaystyle {\begin{aligned}\qquad {\begin{bmatrix}{\tilde {Q}}&{\tilde {S}}\\{\tilde {S}}^{T}&{\tilde {R}}\\\end{bmatrix}}={\begin{bmatrix}Q&S\\S^{T}&R\\\end{bmatrix}}^{-1},U^{\bot }=N([1\quad M^{T}])=R({\begin{bmatrix}-M^{T}\\1\\\end{bmatrix}})\end{aligned}}} , and {\displaystyle {\begin{aligned}V^{\bot }=N([0\quad 1])=R({\begin{bmatrix}1\\0\\\end{bmatrix}})\end{aligned}}}.