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''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" />.

====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>