Understanding Engineering Mathematics

non-trivial solutionsif|A|=0. All this would be of little interest were it not for the
fact that homogeneous systems are very important, especially in engineering mathematics.
When a complicated structure such as an aircraft wing is being studied to examine the
kinds of vibrations it might experience it turns out that the mathematical modelling relies
heavily on homogeneous systems of equations of a particular type that leads to quantities
calledeigenvalues. These quantities are actually related to the frequencies of vibrations
of the various components of the aircraft wing, so their importance is obvious. They
have countless other applications too, but here we will concentrate on their mathematical
significance and properties. Once again, the definitions may be a little complicated and
you may have to take a few things on trust, but the results are worth it, and the technical
manipulations involved are really not that difficult.
Eigenvalues are numerical quantities related to the determinant of a certain type of
square matrix. They arise in, for example, vibrating systems because the mathematical
models used involve systems of equations that take the matrix form

Au=λu

HereAis some square matrix describing the system,λ(Greek, l, ‘lambda’ – the stan-
dard notation for eigenvalues) is some parameter depending on the characteristics of the
vibrating system anduis some ‘solution vector’ that might, for example, represent the
displacements of the vibrating components being modelled. Now this system of equations
is a particular example of a homogeneous system of the form

(A−λI)u= 0

whereIis a unit matrix of the same size asA. It will therefore only have a non-trivial solu-
tion foruif the determinant of coefficients is zero, which gives theeigenvalue equation

|A−λI|= 0

This is in fact a polynomial equation forλ. For a given matrixAonly certain values of
λwill satisfy this equation, and only for these ‘eigenvalues’ will the original equation for
uhave a non-trivial solution. Such non-trivial solutions foruare called theeigenvectors
corresponding to the eigenvaluesλ.
The above is a lot to take in but the objective is to motivate the rather strange definitions
we are introducing. The mechanics of the actual mathematical manipulations are, as noted
above, not too bad. Try the problem quickly!

Problem 13.14

Find the eigenvalues and corresponding eigenvectors of the matrix

A=

[

6 − 3

21

]

Ifλis an eigenvalue ofA, then it satisfies

|A−λI|=

∣

∣

∣

∣

[

6 − 3

21

]

−λ

[

10

01

]∣∣

∣

∣

=

∣

∣

∣

∣

6 −λ − 3

21 −λ

∣

∣

∣

∣