Unitary Matrix

A square matrix is a unitary matrix if

(1)

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

(2)

is a unitary matrix.

Unitary matrices leave the length 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 a unitary basis. That is, each row has length one, and their Hermitian inner product is zero. Similarly, the columns are also a unitary basis. In fact, given any unitary basis, the matrix whose rows are that basis is a unitary matrix. It is automatically the case that the columns are another unitary basis.

A matrix can be tested to see if it is unitary using the Wolfram Language function:

  UnitaryQ[m_List?MatrixQ] :=
    (Conjugate@Transpose @ m . m ==
      IdentityMatrix @ Length @ 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 transpose is a much simpler computation than 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 norm of the determinant of is . Unlike the orthogonal matrices, the unitary matrices are connected. If then is a special unitary matrix.

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

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