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Orthogonal and Unitary Matrices

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A complex $ n\times n$-matrix $ A$ is called unitary if

$\displaystyle A^{-1} = {\overline{A}}^{\operatorname t}=A^\ast\, , $

that is, if the columns of $ A$ form a orthonormal basis of $ \mathbb{C}^n$. For real matrices the complex conjugation can be omitted (as with the scalar product). A unitary matrix with real entries is called orthogonal matrix.

Orthogonal (unitary) matrices form a subgroup of $ \operatorname{GL}(n, \mathbb{R})$ ( $ \operatorname{GL}(n, \mathbb{C})$).

Example:

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  automatically generated 5/23/2011