When the matrix A is symmetric (not, unfortunately, a common occurrence in the equations of flight vehicles) the system is called self-adjoint, and the eigenvectors have the convenient special property of being orthogonal, or normal. That is, the scalar product of any vector with any other is zero, i. e.,
u TUi = u, • иг = 0 Іф j (3.3,20)
In more general cases, when the system is not self-adjoint, and A is an arbitrary n X n matrix, the eigenvectors are neither real nor orthogonal. However, there still exists a reciprocal basis of the eigenvectors, i. e. a set of n vectors Vj orthonormal to the set іц, i. e. such that
Thus the matrix V of the vectors vt evidently satisfies the condition
VrU = I
and clearly YT = U_1
i. e. the columns of V are the rows of U-1. The question now naturally arises as to what system (the adjoint system) has vz – as its eigenvectors, and whether its matrix, В say, has any relation to A. It can be shown that (ref. 3.1) В = AT, i. e. that the matrix of the system adjoint to A is AT and its eigenvectors are orthogonal to those of A.