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Difference between revisions of "Non-derogatory matrix"

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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 $\lambda$. A matrix that is not non-derogatory is said to derogatory.
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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 $\lambda$; 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.
  
 
====References====
 
====References====

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=39804
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article