Jacobian Matrices

Abstract

Before we will be in a position to solve the inverse problem for the systems considered in Section 2.2 we must complete certain mathematical preliminaries. These are related to the special, tridiagonal form of the matrix C in equation (2.2.10). Because of this special form, the eigenvalues and eigenvectors of equation (2.5.3) have special properties, in addition to those described in Chapter 2; the latter are common to all systems with positive definite inertia matrix A and positive semidefinite stiffness matrix C. The most important property of the eigenvalues of such matrices is that they are real and simple, i.e., distinct (Theorem 3.1.3). Thus λ₁ < λ₂ < ... < λ_N. If x^(r) is the rth eigenvector, then, as r increases, the eigenvector oscillates more and more (Theorem 3.3.1) in such a way that the zeros of x^(r) interlace those of the neighbouring x^(r-1) and x^{(r + 1)} (Theorem 3.3.4). We shall now establish these and other results. Throughout Chapters 3 – 5 we make extensive use of the analysis developed by Gantmakher and Krein (1950). Before embarking on the analysis we first reduce the eigenvalue equation

$$(C - \lambda A)x = 0$$

((3.1.1))

to standard form. The matrix A is

$$A = diag({m_1},{m_2},...,{m_N})$$

((3.1.2))

.

Let no one say that I have said nothing new; the arrangement of the subject is new.

Pascal’s Pensées