We consider the class of real second-order linear dynamical systems that admit real diagonal forms with the same eigenvalues and partial multiplicities. The nonzero leading coefficient is allowed to be singular, and the associated quadratic matrix polynomial is assumed to be regular. We present a method and algorithm for converting any such n-dimensional system into a set of n mutually independent second-, first-, and zeroth-order equations. The solutions of these two systems are related by a real, time-dependent, and nonlinear n-dimensional transformation. Explicit formulas for computing the \(2n \times 2n\) real and time-invariant equivalence transformation that enables this conversion are provided. This paper constitutes a complete solution to the problem of diagonalizing a second-order linear system while preserving its associated Jordan canonical form.
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Kawano, D.T., Salsa, R.G. & Ma, F. Decoupling of second-order linear systems by isospectral transformation. Z. Angew. Math. Phys. 69, 137 (2018). https://doi.org/10.1007/s00033-018-1030-x
- Second-order linear differential equations
- Quadratic matrix polynomials
- Isospectral systems
Mathematics Subject Classification