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A NUMERICALINTRODUCTION TOEIGENVALUES USINGMATHEMATICAAllan Alexander Struthers PhD and BSc aa Department of Mathematical Sciences ,Michigan Technological University , Houghton,MI, 49931, USAPublished online: 13 Aug 2007.

To cite this article: Allan Alexander Struthers PhD and BSc (1996) A NUMERICALINTRODUCTION TO EIGENVALUES USING MATHEMATICA , PRIMUS: Problems,Resources, and Issues in Mathematics Undergraduate Studies, 6:1, 21-26, DOI:10.1080/10511979608965805

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Struthers Numerical Introduction to Eigenvalues

A NUMERICAL INTRODUCTION TOEIGENVALUES USING MATHEMATICA

Allan Alexander Struthers

ADDRESS: Department of Mathematical Sciences, Michigan Technologi­cal University, Houghton MI 49931 USA.

ABSTRACT: We offer an introduction to eigenvalues using a form of thepower method is presented.

KEYWORDS: Eigenvalues, eigenvectors, Mathematica, numerical t ech­niques.

INTRODUCTION

An alternative to the traditional, highly algebraic introduction to eigenval­ues is presented. The intent is to motivate the definition

Av=)..v (1)

of the eigenvalues ).. and eigenvectors v of a matrix A using a version ofthe power method. This postpones the algebraic algorithm (find the rootsof the characteristic polynomial etc.) with which the subject is normallyintroduced until the students have gained some experience with the defini­tion. The hope is that the students will retain the definition and gain someappreciation of the utility of numerical techniques. The original motiva­tion was to emphasize the definition rather than the the thicket of algebraiccomputations involved in finding the roots of the characteristic polynomialand solving for the eigenvectors.

At Michigan Technological University, as at many engineering schools,a brief introduction to linear algebra has been squeezed into the calculussequence: between two and three weeks of third quarter calculus is spenton chapters 1-7 of the home-grown supplement by Heuvers et al. [2]. Nu­merous experimental calculus sections have run in the last few years usingdifferent instructional materials and modes of instruction. In school year

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PRIMUS March 1996 Volume VI Number 1

93/94 two sections met five days a week in a computer laboratory using theinteractive calculus text Calculus&Mathematica [1] by Davis, Porta & Uhlwhich consists of a number of Mathematica notebooks. It seemed essentialto incorporate the required linear algebra into the third quarter in a mannerconsistent with the style and format of the text. The introduction to eigen­values described in the paper is contained in the second of the preliminarynotebooks! which were prepared and used in the final three weeks of thirdquarter calculus.

The class consisted of seventeen soon to be sophomores, from a varietyof engineering and science majors who were by this time, most had beenenrolled in the sequence form the start, fairly comfortable with Mathematicaand accustomed to the style of the text. In addition to being able to computethe eigenvalues and eigenvectors of small matrices by hand the students wereexpected to be able to use Mathematica to find the asymptotic behavior ofpowers of matrices and identify the dominant eigenvalue and associatedeigenvector from the asymptotics.

NUMERICAL EXPERIMENTS

The students were asked to guess the behavior of A" for large n with

by experimenting with the Mathematica command N[HatrixPover[A, n]] forvarious values of n: HatrixPover[A, n] computes A" in exact arithmetic; Nforces floating point output. Based on evidence of the form

A 100 = (10)99 [~ :] and A 10 1 = (10)100 [~ :]

the class conjectured that for large n

4 ] = (10)" [ 18 5 2

The pattern is not so clear for

B=[~ ~]------------

1Incompletely edited versions are available from the author by US mail or electroni­cally on the World Wide Web under the URL http://www.math.mtu.edu/home/math/struther/Home.html.

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Struthers

since the same command gives

Numerical Introduction to Eigenvalues

Since exponential growth (which the text [1) has discussed extensively inthe context of population growth and financial planning) is anticipated it isnatural to compute the ratio of successive entries. The command

N[HatrixPOloler[B, n + 1) / HatrixPOloler[B, n)) (3)

does this since, conveniently (if slightly unusually), for Mathematica theratio of matrices is the matrix of ratios of corresponding entries e.g.

With this interpretation (3) gives

BIOI / BIOO = [9.0 9.0]9.0 9.0

which with (2) suggests the conjecture that for large n

The role of the asymptotic directions

can be emphasized by writing the asymptotic formulas in the form

A" "" (10)" [lil: 2a)5 .(9)" _. i'1

and B" "" --[lb: 1bJ.2

(4)

The students correctly guessed the coefficients and denominator in (4) camefrom the asymptotic directions: the coefficients directly; the denominatorsthrough 5 = a.a and 2 = b· b.

With some encouragement the induction argument

A" "" (1~)" [la: u] ==* A"+! "" (1~)" [2Aa: 1Aa]

==* A"+I "" (lO~"+I [U:1a]

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PRIMUS March 1996 Volume VI Number 1

was developed to show the asymptotic formula for A" was consistent. Thestudents recognized the property

Ail..

was key and indicated the direction of ii is unchanged or invariant under A.The class confirmed Bb= 9b and moved onto the larger example

F= [ ~-1

24

-1

-1 ]-1

7 [

8.0 8.0 8.0]for which F l Ol

/ F l OO = 8.0 8.0 8.0 .8.0 8.0 8.0

This suggests 8 for the growth rate and that the asymptotic direction jshould be a solution of F j = s]. The solution

generated the conjecture, rapidly numerically tested, that

F" ....., (8)" [If If _ 21l = (8)" [ ~6 6 -2

11

-2

-2 ]-2

4

To this point there has been no indication that a matrix could havemore than one invariant direction. The students were asked to guess theasymptotic behavior of the remainder term

A" _ (10)" [1 2]5 2 4

for the matrix A. Using similar techniques they rapidly conj ectured that

A" _ (10)" [1 2]....., (-5)" [ 45 2 4 5 -2

-2 ]1 .

They were surprised to find this was an exact relationship and constructedan informal induction proof that

A" = (10)" [1 2] (-5)" [ 45 2 4 + 5 -2

24

-2 ]1 .

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The decomposition

Numerical Introduction to Eigenvalues

n" = (9)" [1 1] (7)" [ 12 1 1 + 2 -1

was r;!'nerated in a similar manner and iterating the process uncovered thethree invariant directions of F and the decomposition

" (8)" [ 1F =-- 16 -2

11

-2

-2] (5)" [ 1-2 +- 14 3 1

111

1] (2)" [ 11 +- -11

20

From these examples the students conjectured that an m x m matrix hall morthogonal invariant directions. They were told this wall true for symmetricmatrices and the ensuing discussion provided a bridge to and motivationfor some careful definitions,

DEFINITIONS

Wit.h this development. as motivation, an eigenvector ii of a matrix M wasdr-fined as an invariant direction of M . The students were reminded that thismr-ant that 1; was a solution of M 1;= >"1; and told that>" was the eigenvaluea<;sociat.!'d with the eigenvector ii. The definition was immediately used tofind t.he !'ip;f'nvalue corresponding to a given eigenvector 1;by computing M ii .The process of finding an eigenvector given an eigenvalue >.. was presentedand the possibility of multiple eigenvectors considered. However, since thefocus was on symmetric matrices the possibility of a deficient matrix wasnot r-mphasized ,

Rf'cOlH:ilinr; their earlier hand computations with the output. from theMathemaiica commands Eigenvalues and Eigenvectors encouraged thestudents to realize that only the direction of an eigenvector is signifi cant.

This prcsentat.ion had the interesting and unanticipated side effect of in­t.roducing a number of more sophisticated concepts from linear algebra. Thefollowing ideas, all contained in Strang's linear algebra text [3], developedfrom student comments and were discussed briefly in class:

• The largest eigenvalue is a measure, called the spectral radius, of thesize of a symmetric matrix and is frequently of interest in applications;

• The spectral radius of A can be estimated by comparing A"+ I t.o A"(a crude version of the power method used for sparse matrices};

• Symmetric matrices decompose into simple matrices formed from!'i~pnvalues and eigenvectors (the spectral theorem) .

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Examinations and quizzes made no mention of these sophisticated con­cepts.

ALGEBRAIC PROCEDURE

The algebraic procedure was discussed in an entirely traditional manner.The characteristic polynomial of M was introduced. An explanation ofwhy the roots of this nth degree polynomial gave the eigenvalues of Mwas presented and the process of finding the eigenvectors from these n, notnecessarily distinct, eigenvalues reviewed. As before the possibility of adeficient matrix was not emphasized.

ACKNOWLEDGMENTS

I would like to acknowledge valuable discussions concerning the presentationof this material with my colleagues William Francis and Sidney Graham.

REFERENCES

1. Davis W., H. Porta, & J. Uhl. 1993. Calculus fj Math ematica. Read­ing MA: Addison-Wesley.

2. Heuvers K., J. Kuisti, W. Francis, G. Ortner, D. Moak, and D. Lock­hart. 1991. Linear Algebra for Calculus. Pacific Grove CA: Brook s/Cole.

3. Strang G. 1988. Linear Algebra and its Applications. Orlando FL:Harcourt Brace Jovanovich Inc.

BIOGRAPHICAL SKETCH

Allan Struthers is an assistant professor of mathematical sciences at Michi­gan Technological University. He received his PhD from Carnegie Mellon in1991 and BSc from the University of Strathclyde (Scotland) in 1985. He is arecipient of the 1995 MTU Distinguished Teaching Award. He is interestedin the applications, particularly those which involve differential equations,of mathematics in science: he is trying to spread this interest by developinga modeling course at MTU.

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