Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.19 EXERCISES

(b) Without assuming thatBis orthogonal, prove thatAis singular.

8.9 Thecommutator[X,Y] of two matrices is defined by the equation

[X,Y]=XY−YX.

Two anticommuting matricesAandBsatisfy

A^2 =I, B^2 =I, [A,B]=2iC.

(a) Prove thatC^2 =Iand that [B,C]=2iA.

(b) Evaluate [[[A,B],[B,C]],[A,B]].

8.10 The four matricesSx,Sy,SzandIare defined by

Sx=

(

01

10

)

, Sy=

(

0 −i

i 0

)

,

Sz=

(

10

0 − 1

)

, I=

(

10

01

)

,

wherei^2 =−1. Show thatS^2 x=IandSxSy=iSz, and obtain similar results

by permuttingx,yandz.Giventhatvis a vector with Cartesian components

(vx,vy,vz), the matrixS(v) is defined as

S(v)=vxSx+vySy+vzSz.

Prove that, for general non-zero vectorsaandb,

S(a)S(b)=a·bI+iS(a×b).

Without further calculation, deduce thatS(a)andS(b) commute if and only ifa

andbare parallel vectors.

8.11 A general triangle has anglesα,βandγand corresponding opposite sidesa,
bandc. Express the length of each side in terms of the lengths of the other
two sides and the relevant cosines, writing the relationships in matrix and vector
form, using the vectors having componentsa, b, cand cosα,cosβ,cosγ. Invert the
matrix and hence deduce the cosine-law expressions involvingα,βandγ.

8.12 Given a matrix

A=





1 α 0

β 10

001



,

whereαandβare non-zero complex numbers, find its eigenvalues and eigenvec-

tors. Find the respective conditions for (a) the eigenvalues to be real and (b) the

eigenvectors to be orthogonal. Show that the conditions are jointly satisfied if

and only ifAis Hermitian.

8.13 Using the Gram–Schmidt procedure:

(a) construct an orthonormal set of vectors from the following:

x 1 =(0011)T, x 2 =(1 0 −10)T,

x 3 =(1202)T, x 4 =(2 1 1 1)T;