A square real matrix has orthonormal columns iff its inverse is its transpose

Created almost 9 years ago, updated about 8 hours ago

For a square real matrix $Q$, orthonormal columns are equivalent to $Q^TQ=I$, which implies $Q^{-1}=Q^T$. For rectangular $Q$ with orthonormal columns, $Q^TQ=I$ still holds but a two-sided inverse need not exist. Then $QQ^T$ is the projection onto its column space.

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