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Unitary matrix

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.

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