Difference between revisions of "Non-derogatory matrix"
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| + | 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=18949
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