Vandermonde and Wronskian matrices over division rings

Lam, T. Y.; Leroy, André

doi:10.1016/0021-8693(88)90063-4

Cited by 119 publications

(211 citation statements)

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“…The first part of the theorem has appeared before in [Li,(4.2)]. Here we offer a shorter, and perhaps also more natural proof, by induction on degA(Z).…”

Section: Theorem 311 For Any Nonzero H E R [E(h A) : C]r < Dega(zmentioning

confidence: 92%

“…We shaU leave the detaüs of this inductive proof to the reader, and present instead a more conceptual proof based on the characterization of the evaluation of a polynomial at a as the remainder of its division by t -a (see [Li,(2.4)]). Using this characterization, we can write p(t) = qx(t)(t -a) + p(a) for some qx(t) e R and g(f) = q2(t')(t' -P(a)) + g(p(a)) for some q2(t) e R'.…”

Section: The Main Formulasmentioning

confidence: 99%

“…We fix a finite Pbasis for A, say {zz,,..., zz"}. By [Li,(2.10)], we know there exists a nonzero polynomial q(t') e K[t',S',LV] vanishing on {p(zzi),... ,p(a")}, with degq(t') < n. Then, by (2.3), (qop)(a¡) = q(p(a¡)) = 0 for z = 1,..., n. Since {ax,... ,a"} form a P-basis for A, this impUes that qop vanishes on A. By (2.3) again, we see that q(p(A)) = 0, so p(A) is (S', LV)-algebraic, with rankp(A) < degqtf) <n = rank A.…”

Section: Preservation Of Algebraic Subsetsmentioning

confidence: 99%

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Hilbert 90 theorems over division rings

Lam¹,

Leroy²

1994

Trans. Amer. Math. Soc.

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Abstract. Hubert's Satz 90 is well-known for cyclic extensions of fields, but attempts at generalizations to the case of division rings have only been partly successful. Jacobson's criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of Satz 90, but the exact relationship between the two was apparently not known. In this paper, we study triples (K,S, D) where S is an endomorphism of the division ring K, and D is an S-derivation. Using the technique of Ore extensions K [t, S, D]. we characterize the notion of (5, D)-algebraicity for elements a e K, and give an effective criterion for two elements a, b € K to be {S, £>)-conjugate, in the case when the (S, Z>)-conjugacy class of a is algebraic. This criterion amounts to a general Hubert 90 Theorem for division rings in the (K ,S, Z))-setting, subsuming and extending all known forms of Hubert 90 in the literature, including the aforementioned Jacobson Criterion. Two of the working tools used in the paper, the Conjugation Theorem (2.2) and the Composite Function Theorem (2.3), are of independent interest in the theory of Ore extensions.

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“…The first part of the theorem has appeared before in [Li,(4.2)]. Here we offer a shorter, and perhaps also more natural proof, by induction on degA(Z).…”

Section: Theorem 311 For Any Nonzero H E R [E(h A) : C]r < Dega(zmentioning

confidence: 92%

Section: The Main Formulasmentioning

confidence: 99%

Section: Preservation Of Algebraic Subsetsmentioning

confidence: 99%

See 1 more Smart Citation

Hilbert 90 theorems over division rings

Lam¹,

Leroy²

1994

Trans. Amer. Math. Soc.

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“…Many mathematicians, such as Cohn [3], Guralnick [11,12], Gustafson [14], Wiegmann [37], Lam and Leroy [17], Hartwig and Putcha [15],Šemrl [25], etc., have made excellent contributions to matrix theory over F . Since, however, the commutative law of multiplication does not hold over F , the results of matrix theory over F have not been more fruitful so far than those over fields, in particular, decompositions of matrices and matrix equations.…”

Section: Introductionmentioning

confidence: 99%

An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications

Wang

Woude

2011

Sci. China Math.

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We in this paper give a decomposition concerning the general matrix triplet over an arbitrary division ring F with the same row or column numbers. We also design a practical algorithm for the decomposition of the matrix triplet. As applications, we present necessary and sufficient conditions for the existence of the general solutions to the system of matrix equationsWe give the expressions of the general solutions to the system and the matrix equation when the solvability conditions are satisfied. Moreover, we present numerical examples to illustrate the results of this paper. We also mention the other applications of the equivalence canonical form, for instance, for the compression of color images. Keywords division ring, linear matrix equation, equivalence canonical form of a matrix triplet, decomposition MSC(2000): 15A21, 15A22, 15A24, 15A33, 15A03 Citation: Wang Q W, van der Woude J W, Yu S W. An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications. Sci China Math, 2011, 54(5): 907-924,

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“…The authors wish to thank the referee for references, in particular [4], and for pointing out a needed modification in Theorem 3.4.…”

Section: Acknowledgementmentioning

confidence: 99%