Inverse eigenvalue problems for damped vibrating systems

First, let us describe what is understood here by a “damped vibrating system.” Consider a system that is determined by n generalized coordinates ql(r),..., q,,(t) and associtated kinetic energy, potential energy, and dis- sipation functions. These are supposed to be homogeneous quadratic forms in the generalized velocities, coordinates, and velocities, respectively, and independent of the time. Motions of such a system are then governed by differential equations of the form Aij+&j+Cq=O, (0.1) where q is the column vector of coordinates, the dots denote derivatives with respect to time, and A, B, C are n x n matrices associated with the respective quadratic forms. In particular, they are generally real and sym- metric with A positive definite and B, C nonnegative (or positive) definite. It is well known that solutions of such a system are intimately connected with the algebraic properties of the matrix-valued function L(i) = A3.2 + BA + C. (0.2) In the terminology of [ 1, 21, L(I) is known as a selfadjoint matrix polynomial. A number L, and a nonzero vector x are called an eigenvalue and associated (right) eigenvector of L(3,) if det L(&)=O and L(&)x=O. Generalized eigenvectors, or Jordan chains (which will be defined shortly) 238