If the above matrices not unitary, they will not be physically realizable, at least for systems with time independent Hamiltonians, which are the only ones being considered here. It must be shown that each of the above operations is unitary. As a reminder, a unitary matrix is one whose inverse is the same as the transpose of its complex conjugate, and unitary matrices represent reversible operators that preserve normalization.

The Walsh-Hadamard transformation is one of the fundamental unitary
transformations used in quantum computing. The proof is simply a
great deal of linear algebra, showing *W*^{2} = *I* (since *W* is real
and symmetric) and is omitted for brevity.

The rotation matrix *R* with
*R*_{ij} = 0 if
*i* *j*, and
*R*_{ii} = *e*^{}. Here is an arbitrary
real number.

It is easy to see *R*'s complex conjugate transposed is the inverse of
*R*. When *R* is multiplied by it's complex conjugate, the only
non-zero elements are on the diagonal, and when the diagonal elements
are multiplied the powers of *e* will cancel, resulting in *e*^{0} = 1
on the diagonal, the identity matrix. [Grover96]

To show the inversion about average matrix *A* is unitary, recall that
*A* may be written as
*A* = - *I* + 2*P* where:

*I*is the identity matrix*P*is the matrix with each element is equal to 1/*N*

Recall that *P*^{2} = *P*.

*A* is real and symmetric, so *A* is its own transposed complex
conjugate, and we must show *A*^{2} = *I*.

*A*^{2} = (- *I* + 2*P*)^{2} = *I*^{2} -2*P* - 2*P* + 4*P*^{2} = *I* - 4*P* + 4*P* = *I*

[Grover96]