## Unitary Matrix

A unitary matrix is a Matrix for which

 (1)

where denotes the Adjoint Operator. This guarantees that
 (2)

Unitary matrices leave the length of a Complex vector unchanged. The product of two unitary matrices is itself unitary. If is unitary, then so is . A Similarity Transformation of a Hermitian Matrix with a unitary matrix gives
 (3)

For Real Matrices, Hermitian is the same as Orthogonal. Unitary matrices are Normal Matrices.

If is a unitary matrix, then the Permanent

 (4)

(Minc 1978, p. 25, Vardi 1991).

References

Arfken, G. Hermitian Matrices, Unitary Matrices.'' §4.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 209-217, 1985.

Vardi, I. Permanents.'' §6.1 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 108 and 110-112, 1991.