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 ${\displaystyle A\in \mathbb {R} ^{m\times n}}$ , Then, ${\displaystyle M_{a}}$  is called a left orthogonal complement of ${\displaystyle A}$  if it satisfies

${\displaystyle M_{a}A=0,\quad {\text{rank}}(M_{a})=m-{\text{rank}}(A)}$ ;

and ${\displaystyle N_{a}}$  is called a right orthogonal complement of ${\displaystyle A}$  if it satisfies

${\displaystyle AN_{a}=0,\quad {\text{rank}}(N_{a})=n-{\text{rank}}(A)}$ .

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

Projection Lemma

Let ${\displaystyle P}$ , ${\displaystyle Q}$  and ${\displaystyle H=H^{T}}$  be given matrices of appropriate dimensions, ${\displaystyle N_{p}}$  and ${\displaystyle N_{q}}$  be the right orthogonal complement of ${\displaystyle P}$  and ${\displaystyle Q}$ , respectively. Then, there exists ${\displaystyle X}$  such that

${\displaystyle H+P^{T}X^{T}Q+Q^{T}XP<0}$

if and only if

${\displaystyle N_{p}^{T}P^{T}=0,\quad N_{q}^{T}Q^{T}=0}$ .