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# Unitary matrix

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A unitary matrix is a square matrix U whose entries are complex numbers and whose inverse is equal to its conjugate transpose U*. This means that
U*U = UU* = I,
where U* is the conjugate-transpose of U and I is the identity matrix.

A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product of two real vectors, thus

<Gx, Gy> = <x, y>,
so also a unitary matrix U satisfies
<Ux, Uy> = <x, y>
for all complex vectors x and y, where <.,.> stands now for the standard inner product on Cn.

A matrix is unitary if and only if its columns form an orthonormal basis of Cn with respect to this inner product.

All eigenvalues of a unitary matrix are complex numbers of absolute value 1, i.e. they lie on the unit circle centered at 0 in the complex plane. The same is true for its determinant.

All unitary matrices are normal, and the spectral theorem therefore applies to them.

All Wikipedia text is available under the terms of the GNU Free Documentation License

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