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,
is the conjugate-transpose of U
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
- <Ux, Uy> = <x, y>
for all complex
, where <.,.> stands now for the standard inner product
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.