Unitary transformation trace invariance

Among the usual advantages of such expressions as Eq. (7-80) and (7-81), one is salient they show forth the invariance of p and w with respect to the choice of the basis functions, u, in terms of which p, a, and P are expressed. The trace, as will be recalled, is invariant against unitary transformations, and the passage from one basis to another is performed by such transformations. The trace is also indifferent to an exchange of the two matrix factors, which is convenient in calculations. Finally, the statistical matrix lends itself to a certain generalization of states from pure cases to mixtures, required in quantum statistics and the theory of measurements we turn to this question in Section 7.9. 

Invariance of the trace of a matrix under unitary transformation corresponds to the invariance of phase density under canonical transformation in classical theory. 

However, as is well known, the trace is invariant with respect to choice of basis functions that are related by a unitary transformation. Thus, rather than working with the basis of eigenfunctions we may, following eqn (4.31), work with respect to the basis of atomic orbitals, to write... 

As a preliminary for deriving a useful expression, we note that the trace of a matrix is invariant under unitary transformation and that Tr(AB) = Tr (BA). For any unitary matrix 11, it follows that 0 = (LflP) = UlP + UUP and that Tr(U W + UPU) = 0, the prime indicating the energy derivative. Therefore, we have... 

Since the trace of a matrix (C = AB) is invariant under a unitary transformation we can write... 

The invariance of symmetry matrices under unitary transformation is the main principle of point group theory. Show 7 M7 =Mdiag, r r=[l] and trace invariance of... 

Furthermore, since the corrections AEso( Z r) may be obtained by diagonalizing Hso and since the trace of a matrix is invariant to unitary transformations, they must satisfy the relation... 

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