Product of Orthogonal Matrix with Transpose is Identity
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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
- 1980: A.J.M. Spencer: Continuum Mechanics ... (previous) ... (next): $2.1$: Matrices: $(2.12)$