Quadratic matrix polynomials with a parameter
Consider a matrix polynomial of the form \(L(\lambda)=A\lambda^ 2+\epsilon B\lambda +C\) where A, B and C are positive definite \(n\times n\) matrices and \(\epsilon\) is a non-negative parameter. Such polynomials arise in a natural way in the study of damped vibrating systems. The present paper examines the behaviour of the (not necessarily real) zeroes of \(d(\lambda,\epsilon)=\det L(\lambda)\) as the parameter \(\epsilon\) varies. First it is shown that for almost all L we have the ''generic case'': d(\(\lambda\),\(\epsilon)\) has at least 2n-1 distinct zeroes in the complex plane for each positive real \(\epsilon\), and it has 2n zeroes except for finitely many \(\epsilon\). It is then shown that the number of exceptional values of \(\epsilon\) always lies between n and \(2n^ 2\). Finally (Theorem 8.2), it is shown that if \(A=I\), and C has n different eigenvalues, then \(L(\lambda)=(I\lambda -Z_ 2(\epsilon))(I\lambda -Z_ 1(\epsilon))\) where \(I\lambda -Z_ 1(\epsilon)\) is the unique analytic continuation of \(I\lambda -iC^{1/2}\). If we suppose that C has been diagonalized, then the entries of \(Z_ 1(\epsilon)\) are analytic functions of the real variable \(\epsilon\) with no poles or branch points, except for possible branch points of multiplicity 2 in the diagonal entries.
scientific article; zbMATH DE number 4010685
scientific article