Unitary Matrix

A unitary matrix is a Matrix ${\hbox{\sf U}}$ for which

$${\hbox{\sf U}}^\dagger ={\hbox{\sf U}}^{-1},$$ (1)

where $\dagger$ denotes the Adjoint Operator. This guarantees that

$${\hbox{\sf U}}^\dagger{\hbox{\sf U}}={\hbox{\sf 1}}.$$ (2)

Unitary matrices leave the length of a Complex vector unchanged. The product of two unitary matrices is itself unitary. If ${\hbox{\sf U}}$ is unitary, then so is ${\hbox{\sf U}}^{-1}$. A Similarity Transformation of a Hermitian Matrix with a unitary matrix gives

$$(uau^{-1})^\dagger = [(ua)(u^{-1})]^\dagger = (u^{-1})^\dagger(ua)^\dagger = (u^\dagger)^\dagger(a^\dagger u^\dagger)$$

$$= uau^\dagger = uau^{-1}.$$ (3)

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

If ${\hbox{\sf M}}$ is a unitary matrix, then the Permanent

$$\vert\mathop{\rm perm}({\hbox{\sf M}})\vert\leq 1$$ (4)

(Minc 1978, p. 25, Vardi 1991).

See also Adjoint Operator, Hermitian Matrix, Normal Matrix, Orthogonal Matrix, Permanent

References

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

Minc, H. Permanents. Reading, MA: Addison-Wesley, 1978.

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