In mathematics, 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`,

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

- <
`G`**x**,`G`**y**> = <**x**,**y**>,

- <
`U`**x**,`U`**y**> = <**x**,**y**>

A matrix is unitary if and only if its columns form an orthonormal basis of **C**^{n} 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.