# LMIs in Control/pages/Projection Lemma

< LMIs in Control | pages

*WIP, Description in progress*

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

## Definition 1: Orthogonal ComplementsEdit

Let , Then, is called a left orthogonal complement of if it satisfies

;

and is called a right orthogonal complement of if it satisfies

.

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

## Projection LemmaEdit

Let , and be given matrices of appropriate dimensions, and be the right orthogonal complement of and , respectively. Then, there exists such that

if and only if

.

*WIP, additional references to be added*

## External LinksEdit

A list of references documenting and validating the LMI.