Embedding polynomial matrices of one variable. (English) Zbl 1175.15019
Summary: A non square matrix with coefficients in \(K[z]\) can (if a condition on its minors is satisfied) be embedded into a square matrix with determinant 1. Finding theoretically and in an algorithmic way an embedding of small degree is solved by a construction with vector bundles on the projective line over \(K\).
MSC:
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 15A54 | Matrices over function rings in one or more variables |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
Keywords:
polynomial matrix; vector bundles on the projective line; Forney sequence; minimal basis; saturated module; embeddingReferences:
| [1] | Beelen, T.; van Dooren, P., A pencil approach for embedding a polynomial matrix into a unimodular matrix, SIAM J. Matrix Anal. Appl., 9, 1, 77-89 (1988) · Zbl 0646.65041 |
| [2] | Eising, R., Polynomial matrices and feedback, IEEE Trans. Automat. Control, 30, 10, 1022-1025 (1985) · Zbl 0565.93013 |
| [3] | Hartshorne, R., Algebraic Geometry, (Grad. Texts in Math., vol. 52 (1977), Springer-Verlag: Springer-Verlag New York-Heidelberg) · Zbl 0532.14001 |
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