Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACES8.3 Using the properties of determinants, solve with a minimum of calculation thefollowing equations for x:x a a 1a x b 1x +2 x +4 x − 3(a)a b x 1=0, (b)x +3 x x+5∣ a b c 1 ∣∣ x − 2 x − 1 x +1 ∣ =0.8.4 Consider the matrices⎛(a) B =⎝ i 0 −i0 −i i−i i 0⎞⎛ √ √ √ ⎞⎠ , (b) C = √ 1 3 −√ 2 − 3⎝ 1 6 −1 ⎠ .82 0 2Are they (i) real, (ii) diagonal, (iii) symmetric, (iv) antisymmetric, (v) singular,(vi) orthogonal, (vii) Hermitian, (viii) anti-Hermitian, (ix) unitary, (x) normal?8.5 By considering the matricesA =(1 00 0) ( )0 0, B =,3 4show that AB = 0 does not imply that either A or B is the zero matrix, but thatit does imply that at least one of them is singular.8.6 This exercise considers a crystal whose unit cell has base vectors that are notnecessarily mutually orthogonal.(a) The basis vectors of the unit cell of a crystal, with the origin O at one corner,are denoted by e 1 , e 2 , e 3 .ThematrixG has elements G ij ,whereG ij = e i · e jand H ij are the elements of the matrix H ≡ G −1 . Show that the vectorsf i = ∑ j H ije j are the reciprocal vectors and that H ij = f i · f j .(b) If the vectors u and v are given byu = ∑ u i e i , v = ∑ v i f i ,iiobtain expressions for |u|, |v|, andu · v.(c) If the basis vectors are each of length a and the angle between each pair isπ/3, write down G and hence obtain H.(d) Calculate (i) the length of the normal from O onto the plane containing thepoints p −1 e 1 , q −1 e 2 , r −1 e 3 , and (ii) the angle between this normal and e 1 .8.7 Prove the following results involving Hermitian matrices:(a) If A is Hermitian and U is unitary then U −1 AU is Hermitian.(b) If A is anti-Hermitian then iA is Hermitian.(c) The product of two Hermitian matrices A and B is Hermitian if and only ifA and B commute.(d) If S is a real antisymmetric matrix then A =(I − S)(I + S) −1 is orthogonal.If A is given by( )cos θ sin θA =− sin θ cos θthen find the matrix S that is needed to express A in the above form.(e) If K is skew-hermitian, i.e. K † = −K, thenV =(I + K)(I − K) −1 is unitary.8.8 A and B are real non-zero 3 × 3 matrices and satisfy the equation(AB) T + B −1 A = 0.(a) Prove that if B is orthogonal then A is antisymmetric.308