Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.16 DIAGONALISATION OF MATRICES

orthonormal and the transformation matrixSis unitary then

〈e′i|e′j〉=

〈∑

k

Skiek

∣

∣

∣

∑

r

Srjer

〉

=

∑

k

Ski∗

∑

r

Srj〈ek|er〉

=

∑

k

Ski∗

∑

r

Srjδkr=

∑

k

Ski∗Skj=(S†S)ij=δij,

showing that the new basis is also orthonormal.

Furthermore, in addition to the properties of general similarity transformations,

for unitary transformations the following hold.

(i) IfAis Hermitian (anti-Hermitian) thenA′is Hermitian (anti-Hermitian),

i.e. ifA†=±Athen

(A′)†=(S†AS)†=S†A†S=±S†AS=±A′. (8.99)

(ii) IfAis unitary (so thatA†=A−^1 )thenA′is unitary, since

(A′)†A′=(S†AS)†(S†AS)=S†A†SS†AS=S†A†AS

=S†IS=I. (8.100)

8.16 Diagonalisation of matrices

Suppose that a linear operatorAis represented in some basisei,i=1, 2 ,...,N,

by the matrixA. Consider a new basisxjgiven by

xj=

∑N

i=1

Sijei,

where thexjare chosen to be the eigenvectors of the linear operatorA,i.e.

Axj=λjxj. (8.101)

In the new basis,A is represented by the matrixA′=S−^1 AS, which has a

particularly simple form, as we shall see shortly. The elementSijofSis theith

component, in the old (unprimed) basis, of thejth eigenvectorxjofA,i.e.the

columns ofSare the eigenvectors of the matrixA:

S=





↑↑ ↑

x^1 x^2 ··· xN

↓↓ ↓



,