LMIs in Control/Tools/Elimination of Variables/Variable Elimination in a Partition Matrix

WIP, Description in progress

Two basic lemmas treat the elimination of variables in a partitioned matrix.

Lemma 1

Consider the matricies $Z=\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{12}^T& Z_{22}\end{array}\right]$, $Z_{11}\in {ℝ}^{n\times n}$, be symmetric. Then, there exists a symmetric matrix $X$ such that

$\left[\begin{array}{ccc}{Z}_{11}-X& Z_{12}& X\\ Z_{12}^T& Z_{22}& 0\\ X& 0& -X\end{array}\right]<0$

if and only if

$Z=\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{12}^T& Z_{22}\end{array}\right]<0$.

Lemma 2

Let $Z_{ij}$, $i=1,2,3,j=i,\cdots ,3$, be given matrices of appropriate dimensions. Then there exists a matrix $X$ such that

$\left[\begin{array}{ccc}{Z}_{11}& Z_{12}& Z_{13}\\ Z_{12}^T& Z_{22}& Z_{23}+X^T\\ Z_{13}^T& Z_{23}^T+X& Z_{33}\end{array}\right]<0$

if and only if

$Z=\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{12}^T& Z_{22}\end{array}\right]<0$, $Z=\left[\begin{array}{cc}{Z}_{11}& Z_{13}\\ Z_{13}^T& Z_{33}\end{array}\right]<0$

In this case, $X$ is given by

$X=Z_{13}^T{Z}_{11}^{-1}{Z}_{12}-Z_{23}^T$.

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.

• [1] - LMI in Control Systems Analysis, Design and Applications