Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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.