Mathematical Methods for Physics and Engineering - Matematica.NET

NORMAL MODESbe shown that they do possess the desirable properties(x j ) T Ax i =0 and (x j ) T Bx i =0 ifi ≠ j. (9.16)This result is proved as follows. From (9.14) it is clear that, for general i and j,(x j ) T (B − ω 2 i A)x i = 0. (9.17)But, by taking the transpose of (9.14) with i replaced by j and recalling that Aand B are real and symmetric, we obtain(x j ) T (B − ω 2 j A) =0.Forming the scalar product of this with x i and subtracting the result from (9.17)gives(ω 2 j − ω 2 i )(x j ) T Ax i =0.Thus, for i ≠ j and non-degenerate eigenvalues ωi2 and ωj 2 , we have that(x j ) T Ax i = 0, and substituting this into (9.17) immediately establishes the correspondingresult for (x j ) T Bx i . Clearly, if either A or B is a multiple of the unitmatrix then the eigenvectors are mutually orthogonal in the normal sense. Theorthogonality 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 |c i| 2 ω 2 i (xi ) T Ax i∑k |c k| 2 (x k ) T Ax k . (9.18)Now, if ω0 2 is the lowest eigenfrequency then ω2 i ≥ ω0 2 for all i and, further, since(x i ) T Ax i ≥ 0 for all i the numerator of (9.18) is ≥ ω02 ∑i |c i| 2 (x i ) T Ax i .Henceλ(x) ≡ xT Bxx T Ax ≥ ω2 0, (9.19)for any x whatsoever (whether x is an eigenvector or not). Thus we are able toestimate the lowest eigenfrequency of the system by evaluating λ for a varietyof vectors x, the components of which, it will be recalled, give the ratios of thecoordinate amplitudes. This is sometimes a useful approach if many coordinatesare involved and direct solution for the eigenvalues is not possible.An additional result is that the maximum eigenfrequency ωm2 may also beestimated. It is obvious that if we replace the statement ‘ωi2 ≥ ω0 2 for all i’ by‘ωi 2 ≤ ωm 2 for all i’, then λ(x) ≤ ω2 m for any x. Thus λ(x) always lies betweenthe lowest and highest eigenfrequencies of the system. Furthermore, λ(x) has astationary value, equal to ωk 2 ,whenx is the kth eigenvector (see subsection 8.17.1).328