Unitary matrix

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

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,

so also a unitary matrix U satisfies

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:

1. A is unitary
2. A^* is unitary
3. the columns of A form an orthonormal basis of Cⁿ with respect to this inner product
4. the rows of A form an orthonormal basis of Cⁿ with respect to this inner product
5. 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.

See also

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