Total decoupling of general quadratic pencils, Part I: Theory

The notion of quadratic pencils, λ 2 M + λ C + K , where M, C, and K are n × n real matrices with or without some additional properties such as symmetry or positive definiteness, plays critical roles in many important applications. It has been long desirable, yet with very limited success, to reduce a complicated high-degree-of-freedom system to some simpler low-degree-of-freedom subsystems. Recently, Garvey, Friswell and Prells proposed a promising approach by which, under some mild assumptions, a general quadratic pencils can be converted by real-valued isospectral transformations into a totally decoupled system. This approach, if numerically feasible, would reduce the original n-degree-of-freedom second-order system to n totally independent single-degree-of-freedom second-order subsystems. Such a claim would be a striking breakthrough in the common knowledge that generally no three matrices M, C, and K can be diagonalized simultaneously. This paper intends to serve three purposes: to clarify some of the ambiguities in the original proposition, to simplify some of the computational details and, most importantly, to complete the theory of existence by matrix polynomial factorization tactics.