Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

MATRICES AND VECTOR SPACES


8.3 Using the properties of determinants, solve with a minimum of calculation the
following equations forx:


(a)


∣∣



∣∣



xaa 1
axb 1
abx 1
abc 1


∣∣



∣∣



=0, (b)



∣∣




x+2 x+4 x− 3
x+3 xx+5
x− 2 x− 1 x+1



∣∣




=0.


8.4 Consider the matrices


(a)B=



0 −ii
i 0 −i
−ii 0


, (b)C=√^1
8




3 −



2 −



3


1



6 − 1


20 2



.


Are 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 matrices


A=

(


10


00


)


, B=


(


00


34


)


,


show thatAB= 0 doesnotimply that eitherAorBis the zero matrix, but that
it 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 not
necessarily mutually orthogonal.
(a) The basis vectors of the unit cell of a crystal, with the originOat one corner,
are denoted bye 1 ,e 2 ,e 3 .ThematrixGhas elementsGij,whereGij=ei·ej
andHijare the elements of the matrixH≡G−^1. Show that the vectors
fi=



jHijejare the reciprocal vectors and thatHij=fi·fj.
(b) If the vectorsuandvare given by

u=


i

uiei, v=


i

vifi,

obtain expressions for|u|,|v|,andu·v.
(c) If the basis vectors are each of lengthaand the angle between each pair is
π/3, write downGand hence obtainH.
(d) Calculate (i) the length of the normal fromOonto the plane containing the
pointsp−^1 e 1 ,q−^1 e 2 ,r−^1 e 3 , and (ii) the angle between this normal ande 1.

8.7 Prove the following results involving Hermitian matrices:


(a) IfAis Hermitian andUis unitary thenU−^1 AUis Hermitian.
(b) IfAis anti-Hermitian theniAis Hermitian.
(c) The product of two Hermitian matricesAandBis Hermitian if and only if
AandBcommute.
(d) IfSis a real antisymmetric matrix thenA=(I−S)(I+S)−^1 is orthogonal.
IfAis given by

A=

(


cosθ sinθ
−sinθ cosθ

)


then find the matrixSthat is needed to expressAin the above form.
(e) IfKis skew-hermitian, i.e.K†=−K,thenV=(I+K)(I−K)−^1 is unitary.

8.8 AandBare real non-zero 3×3 matrices and satisfy the equation


(AB)T+B−^1 A= 0.

(a) Prove that ifBis orthogonal thenAis antisymmetric.
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