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

Solutions of the equation of motion of an n-dimensional vibrating system M q ̈ + D q ̇ + K q = 0 $$M\ddot{q} + D\dot{q} + Kq = 0$$ can be found by solving the quadratic eigenvalue problem L ( λ ) x : = λ 2 M x + λ D x + K x = 0 $$L(\lambda )x:=\lambda ^{2}Mx +\lambda Dx + Kx = 0$$ . 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 Λ = U + i W $$\varLambda = U + iW$$ , 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 Θ ) $$X_{c} = X_{R}(I - i\varTheta )$$ . The identified matrices depend on a matrix polynomial P r ( Θ ) = U r + W r Θ T + Θ W r − Θ W r Θ T , r = − 1 , 0 , 1 $$P_{r}(\varTheta ) = U_{r} + W_{r}\varTheta ^{T} +\varTheta W_{r} -\varTheta W_{r}\varTheta ^{T},r = -1,0,1$$ , where U r = ℜ ( Λ r ) $$U_{r} = \mathfrak{R}(\varLambda ^{r})$$ and W r = ℑ ( Λ r ) $$W_{r} = \mathfrak{I}(\varLambda ^{r})$$ . 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 $$r = -1,1$$ 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.