Normalization of Eigenvectors and Certain Properties of Parameter Matrices Associated with The Inverse Problem for Vibrating Systems

Abstract

Solutions of the equation of motion of an n-dimensional vibrating system $M\ddot{q} + D\dot{q} + Kq = 0$ can be found by solving the quadratic eigenvalue problem $L(\lambda )x:=\lambda ^{2}Mx +\lambda Dx + Kx = 0$. The inverse problem is to identify real definite matrices M > 0, K > 0 and D ≤ 0 from a specified pair (Λ, X _c ) of n-eigenvalues and their corresponding eigenvectors of the eigenvalue problem. We assume here that $\varLambda = U + iW$, where U ≤ 0 and W > 0 are diagonal matrices. The well posedness of the inverse problem requires that the matrix X _c be specially normalized. It is known that for such specially normalized X _c , there exist a nonsingular matrix X _R and an orthogonal matrix Θ, both real, such that $X_{c} = X_{R}(I - i\varTheta )$. The identified matrices depend on a matrix polynomial $P_{r}(\varTheta ) = U_{r} + W_{r}\varTheta ^{T} +\varTheta W_{r} -\varTheta W_{r}\varTheta ^{T},r = -1,0,1$, where $U_{r} = \mathfrak{R}(\varLambda ^{r})$ and $W_{r} = \mathfrak{I}(\varLambda ^{r})$. In this work we give an explicit characterization of normalizers of X _c , introduce some new results on the class of admissible orthogonal matrices Θ and characterize the invertibility of the polynomials P _r (Θ) in terms of the invertibility of Λ ^r. For $r = -1,1$ this is equivalent to identifying M > 0, K > 0. For r = 2 but U _r not strictly negative, we give an example to show that P _r (Θ) is indefinite for all Θ.