If we have an orthogonal square matrix, A, and some c in the complex numbers with |c| = 1, how do we know cA is unitary?
2026-03-26 20:38:33.1774557513
Are orthogonal matrices unitary?
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We know that $AA^T = I $ and $A^TA = I$ (since $A$ is orthogonal). Then, $cA((cA)^\ast)^T = c\overline{c} AA^T = c\overline{c} I = I$ (since $|c| = 1$) (assuming $A$ is real). As also mentioned in the comments, one just has to use the definition of a unitary/orthogonal matrix.