Algorithms for multidimensional spectral factorization and sum of squares

In this paper, algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates, and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems. These algorithms are based on the calculus of 2n-variable polynomial matrices and their associated quadratic differential forms, and share the common feature that the problems are lifted from the original n-variable polynomial context to a 2n-variable polynomial context. This allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities (LMI’s), to the feasibility of a semialgebraic set or to a linear eigenvalue problem.