On the inverse symmetric quadratic eigenvalue problem
The inverse quadratic eigenvalue problem (IQEP) is to find a quadratic matrix polynomial \(L(\lambda)=L_2\lambda^2+L_1\lambda+L_0\) when eigenvalues are (partially) prescribed. The canonical Jordan form, developed in [the first author, ibid. 29, No. 1, 279--301 (2007; Zbl 1132.74017)] and the methods used there are completed in this paper to solve the real symmetric IQEP given the eigenvalues and a sign characteristic. Possibly extra positivity conditions on some coefficients of \(L(\lambda)\) can be prescribed. Essentially the semisimple case is considered, i.e., when algebraic and geometric multiplicities of the eigenvalues coincide. Conditions are given for \(L(\lambda)\) to be diagonalizable. This allows to formulate a procedure to generate not just one, but many different solutions.
scientific article; zbMATH DE number 6333368