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Mathematics
General Math
Matrix of columns of polynomials coefficients invertibility
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[QUOTE="Stephen Tashi, post: 5510959, member: 186655"] One way to interpret your question is "What is the proof that the standard algorithm for decomposing a ratio of polynomials as a sum of fractions works ?". Proofs that algorithms work tend to be tedious and boring, so there may be shortage of volunteers for presenting such a proof. I've seen people say that the key is "Bezout's Lemma" and the rest is left to the reader. If you take for granted that the standard algorithm works then it can't lead to a problem of solving a systems of equations whose associated coefficient matrix is not invertible - so there is a proof-by-contradiction that the rows of such a system are independent (which is a stronger statement than saying the rows are pairwise independent). However, my guess is that you are actually inquiring if there is any interesting mathematics that studies in more detail the relations between invertible matrices and problems of writing a ratio of polynomials as a sum of fractions. If that's what you're curious about, try to formulate some questions that deal with possible relations. I haven't thought about the subject. Just off the top of my head, we could ask questions like: Can any invertible matrix be associated with a matrix that arises in writing some fraction of polynomials as sum of fractions ? Can the relation between such problems and invertible matrices be used to define equivalence classes of invertible matrices that differ from "exact" equality? Can we write matrices whose elements are functions of parameters such that they define a family of invertible matrices ? [/QUOTE]
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Matrix of columns of polynomials coefficients invertibility
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