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 $$Z=\begin{bmatrix} Z_{11} & Z_{12} \\ Z_{12}^T & Z_{22} \end{bmatrix} $$, $$Z_{11}\in\mathbb{R}^{n\times n}$$, be symmetrix. Then, there exists a symmetric matrix $$ X $$ such that

$$ \begin{bmatrix} Z_{11}-X & Z_{12} & X \\ Z_{12}^T & Z_{22} & 0 \\ X & 0 & -X \end{bmatrix} < 0 $$

if and only if

$$Z=\begin{bmatrix} Z_{11} & Z_{12} \\ Z_{12}^T & Z_{22} \end{bmatrix} < 0 $$.

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

$$ \begin{bmatrix} Z_{11} & Z_{12} & Z_{13} \\ Z_{12}^T & Z_{22} & Z_{23}+X^T \\ Z_{13}^T & Z_{23}^T+X & Z_{33} \end{bmatrix} < 0 $$

if and only if

$$Z=\begin{bmatrix} Z_{11} & Z_{12} \\ Z_{12}^T & Z_{22} \end{bmatrix} < 0 $$, $$Z=\begin{bmatrix} Z_{11} & Z_{13} \\ Z_{13}^T & Z_{33} \end{bmatrix} < 0 $$

In this case, $$X$$ is given by

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

WIP, additional references to be added