Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESComparing this with the second equation in (8.93) we find that the componentsof the linear operator A transform asA ′ = S −1 AS. (8.94)Equation (8.94) is an example of a similarity transformation – a transformationthat can be particularly useful in converting matrices into convenient forms forcomputation.Given a square matrix A, we may interpret it as representing a linear operatorA in a given basis e i . From (8.94), however, we may also consider the matrixA ′ = S −1 AS, for any non-singular matrix S, as representing the same linearoperator A but in a new basis e ′ j , related to the old basis bye ′ j = ∑ iS ij e i .Therefore we would expect that any property of the matrix A that representssome (basis-independent) property of the linear operator A will also be sharedby the matrix A ′ . We list these properties below.(i) If A = I then A ′ = I, since, from (8.94),(ii) The value of the determinant is unchanged:A ′ = S −1 IS = S −1 S = I. (8.95)|A ′ | = |S −1 AS| = |S −1 ||A||S| = |A||S −1 ||S| = |A||S −1 S| = |A|. (8.96)(iii) The characteristic determinant and hence the eigenvalues of A ′ are thesame as those of A: from (8.86),|A ′ − λI| = |S −1 AS − λI| = |S −1 (A − λI)S|= |S −1 ||S||A − λI| = |A − λI|. (8.97)(iv) The value of the trace is unchanged: from (8.87),Tr A ′ = ∑ A ′ ii = ∑ ∑ ∑(S −1 ) ij A jk S kiii j k= ∑ ∑ ∑S ki (S −1 ) ij A jk = ∑ ∑δ kj A jk = ∑ A jji j kj kj=TrA. (8.98)An important class of similarity transformations is that for which S is a unitarymatrix; in this case A ′ = S −1 AS = S † AS. Unitary transformation matricesare particularly important, for the following reason. If the original basis e i is284