physics - Unitary matrix

In 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,

$\langle Gx, Gy \rangle = \langle x, y \rangle$

so also a unitary matrix U satisfies

$\langle Ux, Uy \rangle = \langle x, y \rangle$

for all complex vectors x and y, where <.,.> stands now for the standard inner product on Cⁿ. If A is an n by n matrix then the following are all equivalent conditions:

A is unitary
A^* is unitary
the columns of A form an orthonormal basis of Cⁿ with respect to this inner product
the rows of A form an orthonormal basis of Cⁿ with respect to this inner product
A is an isometry with respect to the norm from this inner product

It follows from the isometry property that 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.

A unitary matrix is called special if its determinant is 1.

Unitary matrix

See also