LMIs in Control/pages/Projection Lemma

WIP, Description in progress

A condition for eliminating a variable in an LMI using orthogonal complements is presented.

Definition 1: Orthogonal Complements
Let $$A\in \mathbb{R}^{m\times n} $$, Then, $$M_a$$ is called a left orthogonal complement of $$A$$ if it satisfies

$$ M_aA=0, \quad \text{rank}(M_a)=m-\text{rank}(A) $$;

and $$N_a$$ is called a right orthogonal complement of $$A$$ if it satisfies

$$ AN_a=0, \quad \text{rank}(N_a)=n-\text{rank}(A) $$.

Using the definition of orthogonal complements, we have the following projection lemma:

Projection Lemma
Let $$P$$, $$Q$$ and $$H=H^T$$ be given matrices of appropriate dimensions, $$N_p$$ and $$N_q$$ be the right orthogonal complement of $$P$$ and $$Q$$, respectively. Then, there exists $$X$$ such that

$$H+P^TX^TQ+Q^TXP<0$$

if and only if

$$N_p^TP^T=0, \quad N_q^TQ^T=0 $$.

WIP, additional references to be added