NORMAL MODES
be shown that they do possess the desirable properties
(xj)TAxi=0 and (xj)TBxi=0 ifi=j. (9.16)This result is proved as follows. From (9.14) it is clear that, for generaliandj,(xj)T(B−ωi^2 A)xi= 0. (9.17)But, by taking the transpose of (9.14) withireplaced byjand recalling thatA
andBare real and symmetric, we obtain
(xj)T(B−ω^2 jA)= 0.Forming the scalar product of this withxiand subtracting the result from (9.17)
gives
(ωj^2 −ω^2 i)(xj)TAxi=0.Thus, fori=j and non-degenerate eigenvaluesω^2 i andω^2 j, we have that
(xj)TAxi= 0, and substituting this into (9.17) immediately establishes the corre-
sponding result for (xj)TBxi. Clearly, if eitherAorBis a multiple of the unit
matrix then the eigenvectors are mutually orthogonal in the normal sense. The
orthogonality relations (9.16) are derived again, and extended, in exercise 9.6.
Using the first of the relationships (9.16) to simplify (9.15), we find thatλ(x)=∑
i|ci|(^2) ω 2
i(x
i)TAxi
∑
k|ck|
(^2) (xk)TAxk. (9.18)
Now, ifω^20 is the lowest eigenfrequency thenω^2 i≥ω 02 for alliand, further, since
(xi)TAxi≥0 for allithe numerator of (9.18) is≥ω^20
∑
i|ci|
(^2) (xi)TAxi.Hence
λ(x)≡
xTBx
xTAx
≥ω^20 , (9.19)
for anyxwhatsoever (whetherxis an eigenvector or not). Thus we are able to
estimate the lowest eigenfrequency of the system by evaluatingλfor a variety
of vectorsx, the components of which, it will be recalled, give the ratios of the
coordinate amplitudes. This is sometimes a useful approach if many coordinates
are involved and direct solution for the eigenvalues is not possible.
An additional result is that the maximum eigenfrequencyω^2 mmay also be
estimated. It is obvious that if we replace the statement ‘ω^2 i≥ω 02 for alli’by
‘ωi^2 ≤ωm^2 for alli’, thenλ(x)≤ωm^2 for anyx. Thusλ(x) always lies between
the lowest and highest eigenfrequencies of the system. Furthermore,λ(x) has a
stationaryvalue, equal toωk^2 ,whenxis thekth eigenvector (see subsection 8.17.1).