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An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120090/n1200901.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120090/n1200902.png" /> such that for each of its distinct eigenvalues (cf. [[Eigen value|Eigen value]]; [[Matrix|Matrix]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120090/n1200903.png" /> there is, in its [[Jordan normal form|Jordan normal form]], only one Jordan block with that eigenvalue. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120090/n1200904.png" /> is non-derogatory if and only if its [[Characteristic polynomial|characteristic polynomial]] and minimum polynomial (cf. [[Minimal polynomial of a matrix|Minimal polynomial of a matrix]]) coincide (up to a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120090/n1200905.png" />). 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====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Stoer,  R. Bulirsch,  "Introduction to numerical analysis" , Springer  (1993)  pp. 338ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ch.G. Cullen,  "Matrices and linear transformations" , Dover, reprint  (1990)  pp. 236ff</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Stoer,  R. Bulirsch,  "Introduction to numerical analysis" , Springer  (1993)  pp. 338ff</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  Ch.G. Cullen,  "Matrices and linear transformations" , Dover, reprint  (1990)  pp. 236ff</TD></TR>
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</table>
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[[Category:Special matrices]]

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=18949
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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