Orthogonal complement of a subspace

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

Orthogonal complement of a subspace

Let $S$ be a subspace of $\mathbb{R}^n$ . The orthogonal complement of $S$ , denoted $S^\perp$ , is the subspace of $\mathbb{R}^n$ that contains the vectors orthogonal to all the vectors in $S$ . If the subspace is described as the range of a matrix:

$S = \{ Ax \: : \: x \in \mathbb{R}^n \},$

then the orthogonal complement is the set of vectors orthogonal to the rows of $A$ , which is the nullspace of $A^T$ .

Example: Consider the line in $\mathbb{R}^3$ passing through the origin and generated by the vector $u = (1,2,3)$ . This is a subspace of dimension 1:

$S = \left\{ tu \: : \: t \in \mathbb{R} \right\} = \left\{ \left( \begin{array}{c} t \\ 2t \\ 3t \end{array} \right) \: : \: t \in \mathbb{R} \right\}.$

To find the orthogonal complement, we find the set of vectors that are orthogonal to any vector of the form $tu$ , with arbitrary $t \in \mathbb{R}$ . This is the same set as the set of vectors orthogonal to $u$ itself. So, we solve for $u^T x = 0$ with $x \in \mathbb{R}^3$ :

$x_1 + 2 x_2 + 3x_3 = 0.$

This is equivalent to $x_1 = -2x_2-3x_3$ . This equation characterizes the elements of the orthogonal complement $S^\perp$ , in the sense that any $x \in S^\perp$ can be written as