Difference between revisions of "Non-derogatory matrix"
From Encyclopedia of Mathematics
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| − | An square [[matrix]] $A$ for which the [[characteristic polynomial]] and [[Minimal polynomial of a matrix|minimal polynomial]] coincide (up to a factor $\pm1$). Equivalently, for each of its distinct [[Eigen value|eigenvalue]]s $\lambda$ there is, in the [[Jordan normal form]] for $A$, only one Jordan block with that eigenvalue $\lambda$. | + | An square [[matrix]] $A$ for which the [[characteristic polynomial]] and [[Minimal polynomial of a matrix|minimal polynomial]] coincide (up to a factor $\pm1$). Equivalently, for each of its distinct [[Eigen value|eigenvalue]]s $\lambda$ there is, in the [[Jordan normal form]] for $A$, only one Jordan block with that eigenvalue $\lambda$; this is in turn equivalent to each distinct eigenvalue having only one independent eigenvector, that is, geometric multiplicity one. |
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| + | A '''derogatory matrix''' is one that is not non-derogatory. | ||
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Latest revision as of 22:28, 22 November 2016
2020 Mathematics Subject Classification: Primary: 15A18 [MSN][ZBL]
An square matrix $A$ for which the characteristic polynomial and minimal polynomial coincide (up to a factor $\pm1$). Equivalently, for each of its distinct eigenvalues $\lambda$ there is, in the Jordan normal form for $A$, only one Jordan block with that eigenvalue $\lambda$; this is in turn equivalent to each distinct eigenvalue having only one independent eigenvector, that is, geometric multiplicity one.
A derogatory matrix is one that is not non-derogatory.
References
| [a1] | J. Stoer, R. Bulirsch, "Introduction to numerical analysis" , Springer (1993) pp. 338ff |
| [a2] | Ch.G. Cullen, "Matrices and linear transformations" , Dover, reprint (1990) pp. 236ff |
How to Cite This Entry:
Non-derogatory matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-derogatory_matrix&oldid=39806
Non-derogatory matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-derogatory_matrix&oldid=39806
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article