### Unitary operators

Suppose we have a basis set that is complete. These states might be eigenstates of some hermitian operator, say A. If we have another hermitian operator B, then the eigenstates of B could also serve as a complete basis set, lets call it . We define the operator , and its adjoint . Note that

 (1.46)

and we can show, in the same manner, that . In other words, the adjoint of operator U is its inverse U-1. Operators that have this property are called unitary operators.

Consider any Hilbert space operator where is the matrix representation of X with respect to basis . Alternatively, we can express X in terms of the basis. where is the matrix representation of X in the |bn> basis, or

 (1.47)

We multiply this relation by the bra <al| on the left, and by ket |ap> on the right to get

 (1.48)

but, and or

 (1.49)

In matrix notation this equation is simply written

 (1.50)

that is, operator X, whose matrix representation is given by in the |an> basis, is related to the matrix representation of X in the |bn> basis, , by the above relation. (Note: is the matrix representation of operator X in the |an> representation).

Matrix equations that have this structure are called similarity transformations, and have important properties. On of such properties is that the eigenvalues of a matrix are invariant under a unitary similarity transformation. That means even though , and are different matrices, their eigenvalues are the same since they are related by a unitary similarity transformation. This implies that we can always find for any operator representation so that the transformed matrix is diagonal. It's eigenvalues are then distributed on the diagonal.