EIGENVALUES AND EIGENVECTORS

The roots Ar of the characteristic equations are known as eigenvalues, or characteristic values. Corresponding to each of them is a special set of initial conditions that lead to a specially simple solution in which only one term of (3.3,10) remains, i. e.

у (t) = игеЛг< (a)

where y(0) = ur (b) (3.3,14)

Since the solution of the autonomous system corresponding to a given set of initial conditions is unique, then if (3.3,14a) is a possible solution (and we shall show that it is), then (3.3,146) gives the unique set of initial con­ditions that produce it. The general solution (3.3,10) is seen to be a super­position of these special solutions. ur is the eigenvector corresponding to?.r, and (3.3,14a) is the associated eigenfunction. Substitution of (3.3,14) into (3.3,1) gives

f For a discussion of the practical computation of eM see Appendix D-8 of ref. 3.1.

Since the expansion of (3.3,15) is a set of homogeneous algebraic equations in the unknowns uir a nontrivial solution exists only if the determinant equals zero, i. e. if

|B(Ar)| = 0 (3.3,16)

However, the Xr are the roots of the characteristic equation |B(.s)| = 0, and hence the condition (3.3,16) is automatically met. The vectors ur are then any that satisfy (3.3,15). It should be noted that since the r. h.s. of (3.3,15) is zero, the multiplication of any eigenvector by a scalar produces another eigenvector that has the same “direction” but different magnitude. To find ur we observe that, from the definition of an inverse (3.3,5),

adj В = В-1 |В| (3.3,17)

Premultiplying by В yields

В adj В = ІВІІ = f(s)I (3.3,18)

For any eigenvalue Xr, we have f(Xr) = 0, and hence

B(Ar) adj B(Ar) = 0 (3.3,19)

Since the null matrix has all its columns zero, then it follows that each column of adj B(Ar) is a vector that satisfies (3.3,156). Hence any nonzero column of adj B(Ar) (if there are more than one, they differ only by constant factors) is an eigenvector corresponding to Xr. The eigenvalues and eigen­vectors are the most important properties of autonomous systems. From them one can deduce everything required about its performance and stability. This is illustrated in detail for flight vehicles in Chapter 9.

The n eigenvectors form the eigenmatrix

U = [utu2 • • • u„] = K,] in which ui} is the ith component of the jth vector.