Product of Orthogonal Matrix with Transpose is Identity

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\mathbf Q$ be an orthogonal matrix.

Then:

$\mathbf Q \mathbf Q^\intercal = \mathbf I = \mathbf Q^\intercal \mathbf Q$

where:

$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$
$\mathbf I$ is a unit (identity) matrix


Proof

By definition, an orthogonal matrix is one such that:

$\mathbf Q^\intercal = \mathbf Q^{-1}$

and so the result follows by definition of inverse.

$\blacksquare$


Sources