Namespaces
Variants
Actions

Difference between revisions of "Non-derogatory matrix"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
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.
+
{{TEX|done}}
 +
An $(n\times m)$-matrix $A$ such that for each of its distinct eigenvalues (cf. [[Eigen value|Eigen value]]; [[Matrix|Matrix]]) $\lambda$ there is, in its [[Jordan normal form|Jordan normal form]], only one Jordan block with that eigenvalue. A matrix $A$ 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 $\pm1$). A matrix that is not non-derogatory is said to derogatory.
  
 
====References====
 
====References====
 
<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>
 
<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>

Revision as of 15:16, 1 May 2014

An $(n\times m)$-matrix $A$ such that for each of its distinct eigenvalues (cf. Eigen value; Matrix) $\lambda$ there is, in its Jordan normal form, only one Jordan block with that eigenvalue. A matrix $A$ is non-derogatory if and only if its characteristic polynomial and minimum polynomial (cf. Minimal polynomial of a matrix) coincide (up to a factor $\pm1$). A matrix that is not non-derogatory is said to 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=32057
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article