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