Equivalences for Linearizations of Matrix Polynomials
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Bibliographic record
Abstract
One useful standard method to compute eigenvalues of matrix polynomials P(z)∈ C n x n [z] of degree at most ℓ in z (denoted of grade ℓ, for short) is to first transform P(z) to an equivalent linear matrix polynomial L(z)=zB-A, called a companion pencil, where A and B are usually of larger dimension than P(z) but L(z) is now only of grade 1 in z. The eigenvalues and eigenvectors of L(z) can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of P(z), including those for infinite eigenvalues, can also be recovered from eigenvectors of L(z) if L(z) is what is called a "strong linearization'' of P(z). In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors E(z) and F(z) which, via the equation E(z)L(z)F(z) = diag(P(z), In, …, I_n), explicitly show the equivalence of P(z) and P(z). By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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