COMPUTATIONS FOR A VIBRATING SYSTEM DIAGONALIZE THE VARIANCE

VOL. 18 NO. 2 (1995) 411-415

COMPUTATIONS FOR A VIBRATING SYSTEM DIAGONALIZE THE ^VARIANCE

J. N. BOYDandR N.RAYCHOWDHURY

Department

of Mathematical Sciences Virginia Commonwealth University

Richmond Virginia 23284-2014

(Received August II, 1993 and in revised form September 15, 1993)

ABSTRACT. The transformationstodiagonalize potential energymatrices forcoupledharmonic oscillators willalsodiagonalizethe variance when written in matrixform. After a brief reviewof ageometricalinterpretationof the variance, the transformations are described and anexampleis given.

KEY

WORDS AND PHRASES. Variance, orthogonal and unitary transformations, coupled oscillators,elasticpotential energy.

1991 AMS SubjectClassificationCode. Primary 20C99, Secondary20C35.

INTRODUCTION. Inthis note,wewillbeinterestedin n independentlychosen real numbers

x,,...,x

^{withmean x x} and in their variance

whichcan also bewritten as

()

We

begin

by

recallingageometric interpretation fors

2.

Let

the

n-tuple (x,x2,

^...,x

n)

^{representapoint}

^P

in n-dimensionalEuclideanspace

(E(n))

and let orthogonal unitvectors

al,a

₂

a

be defined in the positive

x,xv...,x,,

directions, respectively. Then thevector

e=xa ^+x2a

^2+...

+xna

^{canbe taken toemanate}

from the origin

O:(0,0 0)

andtoterminateat

P.

Th.e

projection of

e

_{upon the}line

t:x, =x

^{=...=x isgiven bythe innerproduct}

’( al+a2+’’’+an)’ ^qr _nn -11 x

^{assuggestedinFigure 1.}

^In

n-dimensions,the distance

(d)

^from

P

to t satisfiesthe equation

i-I n

Thuss issimply proportional tothesquare ofthe distancefrom

P

to

e. To

pursue the simplification,wefirst rewriteEquation 1 in matrixform:

(2)

Figure 1: TheGeometry of

P,

7,and

e.

Our

problem

istodiagonalize thematrix.

AN EXAMPLE. To

illustrate thecalculations, let us consider the case forn 4. Then 3

x,,x2,x,x

^3s

- ^{(x xxx 4)}

^-1-1

^{-I -I}

^{3 -1-13 -1-1}

- ^{x x x2 xl}

⁴

^{l I}

In E(4),

^weimagine surfaces ofconstant

:(xl,

x2,x,x

4)

^tobe concentric cylinderswith

It

is clearthat inchoosingnew coordinates, wemusttake one coordinatetobe proportionalto

x

⁺

x2

⁺

x

^{+x4 toplaceoneaxis on}

^e.

^{Simplechoices}of the otherthree areproportionalto

x-x

^v

x-x4

^and

-x-x2 +x3 +x

^todetermine the orthogonal transformationmatrix

0 0

0 0

2 2 2 2

1 1

2 2 2 2

(3)

(4)

We

can rewriteEquation4 as

3 -1 -1 -1

xt

-1 3 -1

x2

(xt x2xsx4)$ -IS

^$

4 -1 3 x

-1 -1

00 10

+ + /2

where the new coordinates

x,

x2,x combinetoyieldthesquare ofthedistancefrom which is nowthe x₄ axis. The appearanceofonlythree summands is consistent withthe

4 1 3

degrees

offreedom forthe variance.

ANOTHER APPROACH. Let

us nowreturn toEquation 1. Aftera bit ofalgebra,we seethat the equationfor the variance canberewritten yetagainas

s² _n-1

1_

¹_n

. ^.m (x_x)2. ⁽⁵⁾

HavingobtainedEquation 5, werecognizedit tobe essentially the^{same as}thatfor the elasticpotential

energy

fornmasses symmetrically

arranged

on af’Lxedcircle and

interconnected

by

idealizedharmonicspringsbetween allpairs of masses. Furthermore,we havepreviouslydescribed thissystem.[

1]

Returningtothe case n 4,let us consider the fourmassesonthecircle as shown in Figure2.

Xl

Figure2. The

Coupled

Oscillators.

The particlesareconstrainedtomoveonthecircleandtheirdisplacements from

theirequilibrium positionsare takentobeXl,x2, x3,x_4.

We

choosethe elastic constantsfor all couplingstobe k. Then thepotentialenergy^canbe written as

P.E.

)2 )2 )2

- ^:[<x

⁺

^-,

^/

^{x x}

⁺

^x,)

^/

^.)

⁺

^{, )].}

The symmetrieswhichsimplified the potentialenergy functionwere derived fromthe rotationsofthe circle. The samesymmetriescanbeappliedto

f(xl,

x2, x3,

x4)

^of

Equation4. The four-by-fourmatrix

3 -1 -1 -1 -1 3 -1 -1

canbe diagonalized by^theunitarytransformationhavingmatrixrepresentation -1 -i 1

-i -1

which wascomputedfrom theirreducible representations ofthe

group

ofrotations

by

90*

180

,

²⁷⁰

,

and 360* about thecenterofthe circleof Figure2.

[2,3]

Although

thetransformation carries the coordinates intocomplex numbers, fwhich isproportionaltothe variance remains fixed and real:

3 -1 (x

x4)U

^{I- -1 3}

f(Xl’X2’X"X4) =- ₁₁

^-1₁ ^-I_-1 ^-1₃

^{U_U x2}

x4

40

where Y,,Yi, and r represent the^newnormal coordinates, theircomplex conjugates, and theirmoduli, respectively. Thus thevariancehasagainbeendiagonalized.

In

addition,we now haveunitarymatricestodiagonalizethe variance matrix for all n >2. Theyhave alreadybeencomputedtodiagonalize potential energies.

GENERALIZATION. Thepurpose^{of this noteisto}pointoutthat theunitarytransformations developedfrom thegroupof rotations of the circle enableus tolookatvarianceand

standard deviation.

We

hadpreviouslyusedthese transformations tofindnatural frequencies for oscillatingmass andspringsystems.

We

werepleasedthatpreviouswork solved a newproblemfor us.

We

simply state the generalization. The variance for n _>2 real numbers x,x2,...,x,, canbe computed as

s²

(x)2

^X

1

Theunitarymatrix

n-1 -1 -1 1 x

!i

n-1 -1 1 x2.

-1 -1 n-1

(6)

U= ¹

2hi

4hi

6hi 2(n-1)ni

exp exp-- exp exp 1

4hi 8i 12i

4(n-1)ni

exp exp-- exp exp 1

1 1 1 1 1

which was

developed

fromtherepresentationsof theAbeliangroup of^rotationsthrough

2--En

_of

the circle in ordertodiagonalizea potentialenergymatrix will alsodiagonalizethe variance matrix.

REFERENCES

BOYD, J. N.

and

RAYCHOWDHURY, P.N., A Group

Theoretic

Approach

toGeneralized Harmonic Vibrations in aOneDimensionalLattice,

Int. ^J, of

^Math,

nd

Math.Sci.

9 (1986),

131-136.

BOYD, J. N.

and

RAYCHOWDHURY, P.N., An

ApplicationofProjection

Operators

to aOneDimensionalCrystal, Bull,of

the Inst,

ofMoth,,Academia $inica7

(1979),

133-144.

BOYD, J. N.

and

RAYCHOWDHURY, P.N., Group

Representations in Lagrangian Mechanics,Physica l14A

(1982),

604-608.