188 DETERMINANTS. MATRICES. AÑD GROUP THEORY

where

1-I)~G} 'O>~(D ,nd II>~G)representingstates of spin projection - 1,°, and 1,respectively.Note. Differential operator analogs of these ladder operators appear in Exercise12.6.7.

4.2.21 Vectors A and B are related by the tensor T

B=TA.

Given A and B show that there is no unique solution for the components of T.. ., . ion B/A is undefined (apart from the special case of

en a scalar).

r A-1, an inverse of a given vector A in the sense thatA'A-1=A-1'A=1.

oes not suffice to define A-1 uniquely. A has literally'erses.

4.2.23 If A is diagonal, with all diagonal elements different, and A and B commute,show that B is diagonal.

4.2.24 If A and B are diagonal, show that A and B commute.

4.2.25 Show that trace (ABC) = trace (C BA) ifany two ofthe three matrices commute.

4.2.26 Angular momentum matrices satisfy a commutation relation

[Mi, MJ = iMb i, j, k cyclic.

Show that the trace of each angular momentum matrix vanishes.

4.2.27 (a) The operator Tr replaces a matrix A by its trace; that is,

Tr(A) = trace(A) = ¿aii'

Show that Tr is a linear operator.(b) The operator det replaces a matrix A by its determinant; that is,

det(A) = determinant of A.

Show that det is not a linear operator.

4.2.28 A and B anticommute. Also, A2 = 1, B2 = 1. Show that trace(A) = trace( B) = O.Note. The'Pauli and Dirac (Section 4.5) matrices are specific examples.

4.2.29 With Ix) an N-dimensional column vector and <ylan N-dimensional row vector,show that

trace(lx) <YI)= <ylx).

Note. Ix) <yl means column vector Ix) multiplying row vector <yl. The resultis a square matrix N x N.

4.2.30 (a) If two non singular matrices anticommute, show that the trace of eachone is zero. (Nonsingular means that the determinant ofthe matrix elements4=O.)

- -204

EXERCISES

DETERMINANTS. MATRICES. AND GROUP THEORY

4.3.1

Note. Assume all matrix elements are real.

4.3.2

4.3.3

4.3.4

r

4.3.5

Show that the product of two orthogonal matrices is orthogonal.Note. This is a key step in showing that all n x n orthogonal matrices form agroup (Section 4.10).

If A is orthogonal, show that its determinant has unit magnitude.

If A is orthogonal and det A = + 1, show that aij = Cij' where Cij is the cofactorof aij' This yields the identities of Eg. 1.41 used in Section lA to show that across product ofvectors (in three-space) is itselfa vector.Hint. Note Exercise 4.2.32.

Another set of Euler rotations in common use is1. a rotation about the x3-axis through an angle qJ,counter-

clockwise,2. a rotation about the x'¡-axis through an angle (J,counter-

clockwise, and3. a rotation about the x;-axis through an angle Ij;, counter-

clockwise.If

IX= qJ - n/2

P=O

y = Ij;+ n/2

qJ= IX+ /2

O=p

Ij;= y - n/2,

show that the final systems are identical.

Suppose the Earth is moved (rotated) so that the north paJe goes to 30° north,20° west (original latitude and longitude system) and the 10° west meridianpoints due south.(a) What are the Euler angles describing this rotation?(b) Find the corresponding direction cosines.

(0.9551

ANS. (b) A = 0.00520.2962

-0.2552

0.5221

0.8138

-0.1504

)- 0.8529

0.5000

4.3.7

4.3.8

4.3.9

angle rotation matrix, Eg. 4.87, is invariant under thel.

"p ~ - p,IX+n, y ~ y - n.

Show that the Euler angle rotation matrix A(IX,p, y) satisfies the followingrelations:(a) A-¡(IX,p,y) = A(IX,p, y)(b) A-¡(IX,p, y) = A(-y, -p, -IX).

Show that the trace of the product of a symmetric and an antisymmetric matrixISzero.

Show that the trace of a matrix remains invariant under similarity transforma-tions.

4.3.10 Show that the deterrninant of a matrix remains invariant under similarity trans-formations.

.,........-

EXERCISES 205

Note. These two exercises (4.3.9 and 4.3.10) show that the trace and the determi-nant are independent of the basis. They are characteristics of the matrix (operator)itself.

4.3.11 Show that the property of antisymmetry is invariant under orthogonal similaritytransformations.

4.3.12 A is 2 x 2 and orthogonal. Find the most general form of

A = (a b

)e d'

4.3.13

Compare with two-dimensional rotation.

Ix) and Iy) are column vectors. Under an orthogonal transformation S, Ix') =Slx), Iy') = Si y). Show that the scalar product <xl y) is invariant under thisorthogonal transformation.Note. This is equivalent to the invariance of the dot product of two vectors,Section 1.3.

4.3.14 Show that the sum of the squares of the elements of a matrix remains invariantunder orthogonal similarity transformations.Note. In Exercise 3.7.11 el El - El may be obtained as the sum of the squaresofthe components ofthe matrix (tensor)~v'

4.3.15 As a generalization of Exercise 4.3.14, show that

¿SjkTjk = ¿ SimTim,jk l.m

where the primed and unprimed elements are related by an orthogonal similaritytransformation. This result is useful in deriving invariants in electromagnetictheory (compare Section 3.7).Note. This product Mjk = ¿SjkTjk is sometimes called a Hadamard product.In the framework of tensor analysis, Chapter 3, this exercise becomes a doublecontraction of two second-rank tensors and therefore is clearly a scalar (in-variant) !

4.3.16 A rotation ({J¡+ ({Jlabout the z-axis is carried out as two successive rotations({J¡and ({Jl,each about the z-axis. Use the matrix representation ofthe rotationsto derive the trigonometric identities:

4.3.17

cos( ({J¡+ qJl) = cos qJ¡ COSqJl - sin qJ¡sin qJl

sin(qJ¡ + ({Jl) = sin qJ¡ cos qJl + COSqJ¡ sin qJl'

A column vector V has components V¡ and Vl in an initial (unprimed) system.Calculate V{ and V~for a(a) rotation of the coordinates through an angle of e eounterc/oekwise,(b) rotation ofthe vector through an angle of e c/oekwise.The results for parts (a) and (b) should be identical.

4.3.18 Write a subroutine that will test whether a real N x N matrix is symmetric.Symmetry may be defined as

o s laij - aj;j s ¡;,

where ¡; is somc small tolerance (which allows for truncation error, and so onin the machine).

214 DETERMINANTS, MATRICES, AND GROUP THEORY

Each Eij (exc1usive of the unit matrix) appears in two of the preceding seis. Inaddition to the set of a's, the set of V's has been used extensive1y in re1ativisticquantum theory.

The largest completely commuting seis of Dirac matrices (inc1uding the unitmatrix) have only four matrices.

The discussion of orthogonal matrices in Section 4.3 and unitary matricesin ibis section is only a mere beginning. The further extensions are of vitalconcern in modern "e1ementary" partic1e physics. With the Pauli and Diracmatrices, we can develop spinors for describing electrons, protons, and otherspin ~partic1es. The coordinate system rotations lead to Oj(a, /3,y), the rotationgroup usually represented by matrices in which the elements are functions ofthe Euler angles describing the rotation. The special unitary group SU(3),(composed of 3 x 3 unitary matrices with determinant + 1), has been usedwith considerable success to describe mesons and baryons. These extensionsare considered further in Sections 4.10 to 4.12.

" EXERCISES

4.5.1 Show that

det(A *) = (det A)* = det(A t).

4.5.2 Three angular momentum matrices satisfy the basic commutation relation

[Jx,Jy] = iJz

f índices). If two of the matrices have real elements,the third must be pure imaginary.

hat the trace is positive definite unless S is the null(c) = o.

4.5.5 If A and B are Hermitian matrices, show that (AB + BA) and i(AB - BA)are also Hermitian.

4.5.6 Matrix C is not Hermitian. Show that C + ct and i(C - ct) are Hermitian.This means that a non-Hermitian matrix may be resolved into two Hermitianparts:

C = ~(C + ct) + ~i i(C - ct).

1. al, al, a3, a4, as.

2. V1, Vz, V3, V4, Vs.

3. 61, 6z, 63, P1, Pz.(4.137)

4. al, V1, 61, al, a3.

5. al, Vz, 6z, al, a3.

6. a3, V3, 63, al, al,

---.........

226 DETERMINANTS. MATRICES. AND GROUP THEORY

4.6.6 A has eigenvalues A¡and corresponding eigenvectors Ix). Show thatA-1 has thesame eigenvectors but with eigenvalues A;-I.

4.6.7 A square matrix with zero determinant is labeled singular.(a) If A is singular, show that there is at least one nonzero column vector v

such that

Alv) = O.

(b) If there is a nonzero vector Iv) such that

Alv) = O,

ular matrix. This means that if a matrix (or operator)alue, the mafrix (or operator) has no inverse.

formation diagona1izes each of two matrices. Showmust commute. (This is particularly important in thelation of quantum mechanics.)

4.6.9 Two Hermitian matrices A and B have the same eigenvalues. Show that A andB are related by a unitary similarity transformation.

4.6.10 Find the eigenvalues and an orthonormal (orthogonal and normalized) set ofeigenvectors for the matrices of Exercise 4.2.15.

4.6.11 Show that the inertia matrix for a single particle ofmass m at (x,y,z) has a zerodeterminant. Explain this result in terms of the invariance of the determinantof a matrix under similarity transformations (Exercise 4.3.10) and a possiblerotation of the coordinate system.

.,."

4.6.12 A certain rigid body may be represented by three point masses:

m1=1 at (1,1,-2)

m2=2 at (-1,-1,0)

m3=1 at (1,1,2).

(a) Find the inertia matrix.(b) Diagona1ize the inertia matrix obtaining the eigenvalues and the principal

axes (as orthonorma1 eigenvectors).

4.6.13 z

.....

//

/

(1, O, 1) (III

,(O, 1, 1)III

y/

//

/(1, 1, O)

x

Unit masses are placed as shown in the figure.

234 DETERMINANTS. MATRICES. AND GROUP THEORY

From Eq. 4.203 the Turing estimate ofthe condition number for H4 becomes

KTuring= 4 x 1 x 6480

= 2.59 X 104.

This is a warning that an input error may be multiplied by 25,000 in thecalculation of the output resulto It is a statement that H4 is ill-conditioned.If you encounter a highly ill-conditioned system you have two alternatives(besides abandoning the problem).

a. Try a different mathematical attack.b. Arrange to carry more significant figures and push

through by brute force.

As previously seen, matrix eigenvector-eigenvalue techniques are not limitedto the solution of strictly matrix problems. A further example of the transfer oftechniques from one area to another is seen in the application of matrix tech-niques to the solution of Fredholm eigenvalue integral equations, Section 16.3.In turn, these matrix techniques are strengthened by a variational calculationof Section 17.8.

12matrix has two eigenvectors and corresponding eigen-rs are not necessarily orthogonal. The eigenvalues are not

4.7.2 As an illustration of Exercise 4.7.1, find the eigenvalues and correspondingeigenvectors for

G ~).Note that the eigenvectors are no! orthogonal.

ANS. A¡ = O, r¡ = (2, -1);Az= 4, rz = (2, 1).

4.7.3 If A is a 2 x 2 matrix show that its eigenvalues Asatisfy the equation

Az - Atrace(A) + detA = O.

4.7.4 Assuming a unitary matrix U to satisfy an eigenvalue equation Ur = Ar,showthat the eigenvalues of the unitary matrix have unir magnitude. This same resultholds for real orthogonal matrices.

For n = 4

( 16

-120 240

-l)

-1 -120 1200 - 2700 1680H4 =

-2700 6480 -4200. (4.206)240

-140 1680 -4200 2800