Difference between revisions of "Minimal polynomial of a matrix"
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''minimum polynomial of a matrix'' | ''minimum polynomial of a matrix'' | ||
| − | Let | + | Let $A$ be a [[Matrix|matrix]]. The minimal polynomial of $A$ is the monic [[Polynomial|polynomial]] $g(\lambda)$ of lowest degree such that $g(A)=0$. It divides the [[Characteristic polynomial|characteristic polynomial]] of $A$ and, more generally, it divides every polynomial $f$ such that $f(A)=0$. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Mirsky, "An introduction to linear algebra" , Dover, reprint (1990) pp. 203ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Ch.G. Cullen, "Matrices and linear transformations" , Dover, reprint (1990) pp. 178ff</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Mirsky, "An introduction to linear algebra" , Dover, reprint (1990) pp. 203ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Ch.G. Cullen, "Matrices and linear transformations" , Dover, reprint (1990) pp. 178ff</TD></TR></table> | ||
Latest revision as of 15:17, 1 May 2014
minimum polynomial of a matrix
Let $A$ be a matrix. The minimal polynomial of $A$ is the monic polynomial $g(\lambda)$ of lowest degree such that $g(A)=0$. It divides the characteristic polynomial of $A$ and, more generally, it divides every polynomial $f$ such that $f(A)=0$.
References
| [a1] | L. Mirsky, "An introduction to linear algebra" , Dover, reprint (1990) pp. 203ff |
| [a2] | Ch.G. Cullen, "Matrices and linear transformations" , Dover, reprint (1990) pp. 178ff |
How to Cite This Entry:
Minimal polynomial of a matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_polynomial_of_a_matrix&oldid=16850
Minimal polynomial of a matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_polynomial_of_a_matrix&oldid=16850
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article