Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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MATRICES AND VECTOR SPACES


8.14 Determination of eigenvalues and eigenvectors

The next step is to show how the eigenvalues and eigenvectors of a givenN×N


matrixAare found. To do this we refer to (8.68) and as in (8.69) rewrite it as


Ax−λIx=(A−λI)x= 0. (8.85)

The slight rearrangement used here is to writexasIx,whereIis the unit matrix


of orderN. The point of doing this is immediate since (8.85) now has the form


of a homogeneous set of simultaneous equations, the theory of which will be


developed in section 8.18. What will be proved there is that the equationBx= 0


only has a non-trivial solutionxif|B|= 0. Correspondingly, therefore, we must


have in the present case that


|A−λI|= 0 , (8.86)

if there are to be non-zero solutionsxto (8.85).


Equation (8.86) is known as thecharacteristic equationforAand its LHS as

thecharacteristicorsecular determinantofA. The equation is a polynomial of


degreeNin the quantityλ.TheNroots of this equationλi,i=1, 2 ,...,N, give


the eigenvalues ofA. Corresponding to eachλithere will be a column vectorxi,


which is theith eigenvector ofAand can be found by using (8.68).


It will be observed that when (8.86) is written out as a polynomial equation in

λ, the coefficient of−λN−^1 in the equation will be simplyA 11 +A 22 +···+ANN


relative to the coefficient ofλN. As discussed in section 8.8, the quantity


∑N
i=1Aii
is thetraceofAand, from the ordinary theory of polynomial equations, will be


equal to the sum of the roots of (8.86):


∑N

i=1

λi=TrA. (8.87)

This can be used as one check that a computation of the eigenvaluesλihas been


done correctly. Unless equation (8.87) is satisfied by a computed set of eigenvalues,


they have not been calculated correctly. However, that equation (8.87) is satisfied is


a necessary, but not sufficient, condition for a correct computation. An alternative


proof of (8.87) is given in section 8.16.


Find the eigenvalues and normalised eigenvectors of the real symmetric matrix

A=




11 3


11 − 3


3 − 3 − 3



.


Using (8.86),


∣∣


1 −λ 13
11 −λ − 3
3 − 3 − 3 −λ



∣∣




=0.

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