Computing Minimal Polynomials of Matrices¶
Neunhöffer, M., & Praeger, C. E. (2008). Computing Minimal Polynomials of Matrices. LMS Journal of Computation and Mathematics.
Cited by¶
1 citation across 1 artifact.
Each citation links to the sentence it supports in the citing article.
Domain-specific¶
- Minimal Polynomial (Linear Algebra)
- . Matrix functions and recurrences. Powers and polynomial expressions reduce modulo \(m_T\), turning long sequences of matrix powers into recurrences of order \(\deg m_T\). Krylov and cyclic-subspace methods. Linear dependence among \(v,Tv,T^2v,\ldots\) yields a vector-relative order polynomial; local polynomials are combined to determine the operator invariant. Computational algebra over finite fields. Randomized and deterministic algorithms compute and verify minimal polynomials for large matrices, canonical-form work, module structure, and matrix-group calculations
This sourceGives an O(n^3) Monte Carlo algorithm and an accompanying deterministic verification procedure for minimal polynomials of matrices over finite fields, motivated by Frobenius normal form, F[x]-module structure and matrix group recognition. The paper's own mechanism: order polynomials of random vectors under the matrix action are combined by least common multiple to reach the minimal polynomial with high probability, and a deterministic algorithm then certifies or refutes the candidate.
SupportedVerified against the source
- . Matrix functions and recurrences. Powers and polynomial expressions reduce modulo \(m_T\), turning long sequences of matrix powers into recurrences of order \(\deg m_T\). Krylov and cyclic-subspace methods. Linear dependence among \(v,Tv,T^2v,\ldots\) yields a vector-relative order polynomial; local polynomials are combined to determine the operator invariant. Computational algebra over finite fields. Randomized and deterministic algorithms compute and verify minimal polynomials for large matrices, canonical-form work, module structure, and matrix-group calculations
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