Unitary Matrix

A unitary matrix is a square matrix of size with complex numbers as entries, satisfying

(1)

where denotes the conjugate transpose and is the matrix inverse. For example,

(2)

is a unitary matrix.

Unitary matrices leave the Euclidean norm of a complex vector unchanged.

For real matrices, unitary is the same as orthogonal. In fact, there are some similarities between orthogonal matrices and unitary matrices. The rows of a unitary matrix are an orthonormal basis. That is, each row has Euclidean norm one, and the Hermitian inner product of any two distinct rows is zero. Similarly, the columns are also an orthonormal basis. In fact, given any orthonormal basis of the complex vector space , the matrix whose rows are the vectors of that orthonormal basis is a unitary matrix. Its columns are another orthonormal basis.

A square matrix can be tested to see if it is unitary in the Wolfram Language using UnitaryMatrixQ[m].

The definition of a unitary matrix guarantees that

(3)

where is the identity matrix. In particular, a unitary matrix is always invertible, and . Note that conjugate transpose is a much simpler computation than matrix inverse. A similarity transformation of a Hermitian matrix with a unitary matrix gives

			(4)
			(5)
			(6)
			(7)
			(8)

Unitary matrices are normal matrices. If is a unitary matrix, then the permanent

(9)

(Minc 1978, p. 25; Vardi 1991).

The unitary matrices are precisely those matrices which preserve the Hermitian inner product

(10)

Also, the complex modulus of the determinant of is . Unlike the orthogonal matrices, the unitary matrices are connected. If then is a special unitary matrix.

The matrix product of two unitary matrices of the same size is another unitary matrix. The matrix inverse of a unitary matrix is another unitary matrix, and identity matrices are unitary. Hence the set of unitary matrices forms a group, called the unitary group.