# LMi for Minimizing Eigenvalues of a Matrix

LMi for Minimizing Eigenvalues of a Matrix

## The System

{\displaystyle {\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}}

Note that {\displaystyle {\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}} are symmetric matrices.

## The Data

{\displaystyle {\begin{aligned}A_{0},A_{1},...,A_{n}\quad {\text{are given matrices.}}\end{aligned}}}

## The Optimization Problem

Find

{\displaystyle {\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}}

to minimize,

{\displaystyle {\begin{aligned}J(x)=\lambda _{max}(A(x))\end{aligned}}}

According to Lemma 1.1 in [1] page 10, the following statements are equivalent

{\displaystyle {\begin{aligned}\lambda _{max}(A(x))\leq t\iff A(x)-tI\leq 0\end{aligned}}}

## The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.

{\displaystyle {\begin{aligned}{\text{min}}\;\quad t:&\\{\text{s.t.}}\quad {\begin{bmatrix}A(x)-tI\end{bmatrix}}&\leq 0\end{aligned}}}

## Conclusion:

{\displaystyle {\begin{aligned}x_{i},\quad i=1,2,...,n\quad {\text{and}}\quad t>0\end{aligned}}} are parameters to be optimized

## Implementation

A link to Matlab codes for this problem in the Github repository:

## Related LMIs

LMI for Matrix Norm Minimization

LMI for Schur Stabilization