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The Minimal polynomials, characteristic subspaces, normal bases and the frobenius form

Paul Camion 1 Daniel Augot 1
1 CODES - Coding and cryptography
Inria Paris-Rocquencourt
Abstract : Various algorithms connected with the computation of the minimal polynomial of a square n x n matrix over a field k are presented here. Only classical arithmetic is used. The complexity is evaluated only for k = Fq. The complexity of the first algorithm, where the complete factorization of the characteristic polynomial is needed, is[??][??] It produces the minimal polynomial and all characteristic subspaces. Using the shift-Hessenberg form, known to automation scientists and which exists for any matrix, the complexity of this algorithm is reduced to [??] is a parameter for the matrix A, expected to be low. Furthermore an iterative algorithm for the minimal polynomial is presented with complexity [??] where m is a parameter of the used shift-Hessenberg matrix. It does not require knowledge of the characteristic polynomial. A refinement leads to an algorithm where up to m processes can be done independently in parallel. Important here is the fact that the average value of m or mA is E log n. Next we are concerned with the topic of finding a cyclic vector first for a matrix whose characteristic polynomial is square-free. Using the shift-Hessenberg form leads to an algorithm at cost [??] A more sophisticated recurrent procedure gives the result in [??] steps. In particular, a normal basis for an extended finite field will be obtained with that complexity with a deterministic algorithm and with a probabilistic algorithm as well on the date of a matrix representing the Frobenius operator. Finally the Frobenius form is obtained with asymptotic average complexity [??]As a by-product we there obtain a cyclic vector for any matrix. All algorithms are deterministic. In all four cases, the complexity obtained is better than for the heretofore best known deterministic algorithm. The asymptotic expected value of m or mA is log n. The results are summarized in tables 1, 2, 3 and 4. Studying basic properties of the shift-Hessenberg form leads to an algorithm to construct any element in GL(n, k) or may be in a particular subgroup of GL(n, k), of the centralizer for any given matrix. That investigation into the centralizer lead us to extend a result obtained by R. Stong, which allows the needed complexity evaluations to be established and also the size of the centralizer of a linear operator to be given explicitly, for k a finite field, after computation of what is here called an expanded-Frobenius form.
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Submitted on : Wednesday, May 24, 2006 - 4:02:20 PM
Last modification on : Thursday, February 3, 2022 - 11:18:24 AM
Long-term archiving on: : Tuesday, April 12, 2011 - 6:12:09 PM


  • HAL Id : inria-00074666, version 1



Paul Camion, Daniel Augot. The Minimal polynomials, characteristic subspaces, normal bases and the frobenius form. [Research Report] RR-2006, INRIA. 1993. ⟨inria-00074666⟩



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