Orthogonal matrix and projector

A matrix $Q\in {\mathbb{R}}^{m\times n}$ is orthogonal if

$$Q^{T}Q=I$$

This means that the columns of $Q$ are orthogonal to each other.

Notation: ${}^H$ : transpose conjugate

Unitary: same for complex matrices: $Q^{H}Q=I$.

Orthogonal matrices are very important in numerical linear algebra. They often guarantee that an algorithm will be accurate.

Orthogonal matrices are isometries, that is, they conserve the 2-norm of a vector:

$$\Vert Qx{\Vert }_2=\Vert x{\Vert }_2$$

If $Q$ is square, then its inverse is $Q^T$. In that case, $Q$ can be decomposed into a sequence of rotations and reflections.

Fun fact: Cartan–Dieudonné theorem. Every orthogonal transformation in ${\mathbb{C}}^n$ can be described as the composition of at most $n$ reflections.

The matrix $Q{Q}^T$ is not equal to $I$ if $Q$ is rectangular (i.e., $m>n$).

The matrix $P=Q{Q}^T$ represents an orthogonal projection onto $R(Q)$ along $N(Q^T)$.

Dot product, Vector norms, Operator and matrix norms, Projection