Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

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

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