Mathematical Methods for Physics and Engineering - Matematica.NET

8.13 EIGENVECTORS AND EIGENVALUESeigenvector of A corresponding to eigenvalue λ i . But the eigenvector solutions of(A − λ i I)x i = 0 are unique to within a scale factor, and we therefore conclude thatBx i = µ i x ifor some scale factor µ i . However, this is just an eigenvector equation for B andshows that x i is an eigenvector of B, in addition to being an eigenvector of A. Byreversing the roles of A and B, it also follows that every eigenvector of B is aneigenvector of A. Thus the two sets of eigenvectors are identical.(ii) Now suppose that A and B have all their eigenvectors in common, a typicalone x i satisfying bothAx i = λ i x i and Bx i = µ i x i .As the eigenvectors span the N-dimensional vector space, any arbitrary vector xin the space can be written as a linear combination of the eigenvectors,i=1x =N∑c i x i .Now consider bothN∑N∑N∑ABx = AB c i x i = A c i µ i x i = c i λ i µ i x i ,andi=1i=1i=1i=1i=1N∑N∑N∑BAx = BA c i x i = B c i λ i x i = c i µ i λ i x i .It follows that ABx and BAx are the same for any arbitrary x and hence that(AB − BA)x = 0for all x. Thatis,A and B commute.This completes the proof that a necessary and sufficient condition for twonormal matrices to have a set of eigenvectors in common is that they commute.It should be noted that if an eigenvalue of A, say, is degenerate then not all ofits possible sets of eigenvectors will also constitute a set of eigenvectors of B.However, provided that by taking linear combinations one set of joint eigenvectorscan be found, the proof is still valid and the result still holds.When extended to the case of Hermitian operators and continuous eigenfunctions(sections 17.2 and 17.3) the connection between commuting matrices anda set of common eigenvectors plays a fundamental role in the postulatory basisof quantum mechanics. It draws the distinction between commuting and noncommutingobservables and sets limits on how much information about a systemcan be known, even in principle, at any one time.279i=1