Unitary matrix

mathematics, a unitary matrix is a n by n complex matrix U satisfying the condition

U^*U = UU^* = I_n

where I_n is the identity matrix and U^* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse which is equal to its conjugate transpose U^*.

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,

<math>\langle Gx, Gy \rangle = \langle x, y \rangle<math>

so also a unitary matrix U satisfies

<math>\langle Ux, Uy \rangle = \langle x, y \rangle<math>

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 the determinant.

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

See also

• orthogonal matrix
• symplectic matrix
• unitary group
• unitary operator