Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

a result any arbitrary vectorycan be expressed as a linear combination of the

eigenvectorsxi:

y=

∑N

i=1

aixi, (8.79)

whereai=(xi)†y. Thus, the eigenvectors form an orthogonal basis for the vector

space. By normalising the eigenvectors so that (xi)†xi= 1 this basis is made

orthonormal.

Show that a normal matrixAcan be written in terms of its eigenvaluesλiand orthonormal

eigenvectorsxias

A=

∑N

i=1

λixi(xi)†. (8.80)

The key to proving the validity of (8.80) is to show that both sides of the expression give
thesameresultwhenactingonanarbitaryvectory.SinceAis normal, we may expandy
in terms of the eigenvectorsxi, as shown in (8.79). Thus, we have

Ay=A

∑N

i=1

aixi=

∑N

i=1

aiλixi.

Alternatively, the action of the RHS of (8.80) onyis given by

∑N

i=1

λixi(xi)†y=

∑N

i=1

aiλixi,

sinceai=(xi)†y. We see that the two expressions for the action of each side of (8.80) ony
are identical, which implies that this relationship is indeed correct.

8.13.2 Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices

For a normal matrix we showed that ifAx=λxthenA†x=λ∗x. However, ifAis

also Hermitian,A=A†, it follows necessarily thatλ=λ∗. Thus, the eigenvalues

of an Hermitian matrix are real, a result which may be proved directly.

Prove that the eigenvalues of an Hermitian matrix are real.

For any particular eigenvectorxi, we take the Hermitian conjugate ofAxi=λixito give

(xi)†A†=λ∗i(xi)†. (8.81)

UsingA†=A,sinceAis Hermitian, and multiplying on the right byxi,weobtain

(xi)†Axi=λ∗i(xi)†xi. (8.82)

But multiplyingAxi=λixithrough on the left by (xi)†gives

(xi)†Axi=λi(xi)†xi.

Subtracting this from (8.82) yields

0 =(λ∗i−λi)(xi)†xi.