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Fundamentals of Systems of Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
We saw in Section 5.4, that being able to write A in its eigencoordinates greatly simplified our solution to the differential equation x′ = Ax. In doing so, we used ui to represent the eigencoordinates, solved the system u' = Λu for u, and converted back to our original variable via x = Vu. We will be able to state a similar result for matrices that are written in Jordan Canonical Form (sometimes called Jordan Normal Form).
Multidimensional realisation theory and polynomial system solving
Published in International Journal of Control, 2018
Philippe Dreesen, Kim Batselier, Bart De Moor
Let us briefly discuss the case of multiple roots. For a system of multivariate polynomials, a μ-fold solution gives rise to a null space spanned by linear combinations of vectors of the form where the factor (α1!⋅⋅⋅αn!)−1 serves as a normalisation. For a thorough treatment of the so-called dual space of , we refer to Batselier, Dreesen, and De Moor (2014b), and Dayton, Li, and Zeng (2011). The shift relation in the Vandermonde basis involves in the case of multiple roots a Jordan-like normal form where is uppertriangular with zi, evaluated at the m roots, on the diagonal. Some uppertriangular elements of are nonzero, which can be analysed by inspection of . In the same way as for the one-dimensional case, the occurrence of a Jordan normal form gives rise to so-called Jordan chains in the state-space realisation. In practice, the computation of the Jordan normal form is numerically ill-posed, and can be avoided by computing a Schur decomposition (Batselier et al., 2014b).