Commutativity of spectral divisors of quadratic pencils

The object of study of the presented work are quadratic matrix pencils, in other words, quadratic functions of a complex variable, the coefficients of which being the Hermitian matrices. Such functions naturally appear when different problems of mechanics, geophysics or engineering are being studied. In particular, when an oscillatory system of mass-strings with dampers is described, the pencil coefficients characterize the stiffness of the springs and the set dampers. In this context, the so called inverse problems for matrix pencils are of special interest, that is, the problems of building pencils with preset properties. In our work, we study the possibility of building quadratic pencils, allowing for commuting linear factorization. It is well known that any quadratic pencil may be presented as a product of linear (not necessarily commuting) factors, which are called spectral divisors. Henceforth, the description of the structure of one spectral divisor through the structure of another has been studied in several works of the recent decade. We have obtained the criterion, which describes a set of spectral divisors, for each one of which there exists a second commuting spectral divisor. For each element of this set, a structure of all spectral divisors, commuting with it, is described. The criterion of uniqueness of solution to this problem is given. We should note that the conditions of this criterion can be verified for any set quadratic matrix. The obtained results allow to build quadratic pencils, allowing for commuting spectral factorization. Without loss of generality, it is assumed that the left spectral divisor is set. The case when the right spectral divisor is set, is reduced to the considered situation through conjugation operation.