Unitary matrices

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A set of $n$ vectors in $\mathbb C^n$ is orthogonal if it is so with respect to the standard complex scalar product, and orthonormal if in addition each vector has norm 1. Similarly, one has the complex analogue of a matrix being orthogonal.

An $n\times n$ complex matrix $U$ is unitary if $U^**U = I$ , or equivalently if $U^{-1} = U^*$ .

Just as orthogonal matrices are exactly those that preserve the dot product, we have

A complex $n\times n$ matrix is unitary iff ${\bf w}^**{\bf v} = {\bf v}\cdot {\bf w} = (U*{\bf v})\cdot (U*{\bf w})\qquad \forall {\bf v},{\bf w}\in \mathbb C^n$

Proof: Essentially the same as in the real case; by Theorem thm:matrepcomp of the previous section we see that the hypothesis on $U$ implies $U^**U = I^**I = I$ . $\blacksquare$

Unitary matrices are special examples of linear transformations which preserve Hermitian inner products. More on this below.