Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

Comparing this with the second equation in (8.93) we find that the components

of the linear operatorAtransform as

A′=S−^1 AS. (8.94)

Equation (8.94) is an example of asimilarity transformation– a transformation

that can be particularly useful in converting matrices into convenient forms for

computation.

Given a square matrixA, we may interpret it as representing a linear operator

Ain a given basisei. From (8.94), however, we may also consider the matrix

A′=S−^1 AS, for any non-singular matrixS, as representing the same linear

operatorAbut in a new basise′j, related to the old basis by

e′j=

∑

i

Sijei.

Therefore we would expect that any property of the matrixAthat represents

some (basis-independent) property of the linear operatorAwill also be shared

by the matrixA′. We list these properties below.

(i) IfA=IthenA′=I, since, from (8.94),

A′=S−^1 IS=S−^1 S=I. (8.95)

(ii) The value of the determinant is unchanged:

|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 ofA′are the

same as those ofA: 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),

TrA′=

∑

i

A′ii=

∑

i

∑

j

∑

k

(S−^1 )ijAjkSki

=

∑

i

∑

j

∑

k

Ski(S−^1 )ijAjk=

∑

j

∑

k

δkjAjk=

∑

j

Ajj

=TrA. (8.98)

An important class of similarity transformations is that for whichSis a uni-

tary matrix; in this caseA′=S−^1 AS=S†AS. Unitary transformation matrices

are particularly important, for the following reason. If the original basiseiis