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Difference between revisions of "Minimal polynomial of a matrix"

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''minimum polynomial of a matrix''
 
''minimum polynomial of a matrix''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120200/m1202001.png" /> be a [[Matrix|matrix]]. The minimal polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120200/m1202002.png" /> is the monic [[Polynomial|polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120200/m1202003.png" /> of lowest degree such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120200/m1202004.png" />. It divides the [[Characteristic polynomial|characteristic polynomial]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120200/m1202005.png" /> and, more generally, it divides every polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120200/m1202006.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120200/m1202007.png" />.
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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=32058
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article