Mathematical Methods for Physics and Engineering - Matematica.NET

8.19 EXERCISES(b) Without assuming that B is orthogonal, prove that A is singular.8.9 The commutator [X, Y] of two matrices is defined by the equation[X, Y] =XY − YX.Two anticommuting matrices A and B satisfyA 2 = I, B 2 = I, [A, B] =2iC.(a) Prove that C 2 = I and that [B, C] =2iA.(b) Evaluate [[[A, B], [B, C]], [A, B]].8.10 The four matrices S x , S y , S z and I are defined by( ) ( )0 10 −iS x =, S1 0y =,i 0( ) ( )1 01 0S z =, I =,0 −10 1where i 2 = −1. Show that S 2 x = I and S x S y = iS z , and obtain similar resultsby permutting x, y and z. Giventhatv is a vector with Cartesian components(v x ,v y ,v z ), the matrix S(v) is defined asS(v) =v x S x + v y S y + v z S z .Prove that, for general non-zero vectors a and b,S(a)S(b) =a · b I + i S(a × b).Without further calculation, deduce that S(a) andS(b) commute if and only if aand b are parallel vectors.8.11 A general triangle has angles α, β and γ and corresponding opposite sides a,b and c. Express the length of each side in terms of the lengths of the othertwo sides and the relevant cosines, writing the relationships in matrix and vectorform, using the vectors having components a, b, c and cos α, cos β,cos γ. Invert thematrix and hence deduce the cosine-law expressions involving α, β and γ.8.12 Given a matrixA =⎛⎝ 1 β α 1 00⎞⎠ ,0 0 1where α and β are non-zero complex numbers, find its eigenvalues and eigenvectors.Find the respective conditions for (a) the eigenvalues to be real and (b) theeigenvectors to be orthogonal. Show that the conditions are jointly satisfied ifand only if A is Hermitian.8.13 Using the Gram–Schmidt procedure:(a) construct an orthonormal set of vectors from the following:x 1 = (0 0 1 1) T , x 2 =(1 0 − 1 0) T ,x 3 = (1 2 0 2) T , x 4 =(2 1 1 1) T ;309