View source for Semi-simple matrix

A square [[Matrix|matrix]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843801.png" /> similar to a matrix of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843802.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843803.png" /> is a matrix over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843804.png" /> whose characteristic polynomial is irreducible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843806.png" /> (cf. [[Irreducible polynomial|Irreducible polynomial]]). For a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843807.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843808.png" />, the following three statements are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s0843809.png" /> is semi-simple; 2) the minimum polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438010.png" /> has no multiple factors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438011.png" />; and 3) the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438012.png" /> is semi-simple (cf. [[Semi-simple algebra|Semi-simple algebra]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438013.png" /> is a [[Perfect field|perfect field]], then a semi-simple matrix over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438014.png" /> is similar to a diagonal matrix over a certain extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438015.png" />. For any square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438016.png" /> over a perfect field there is a unique representation in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438018.png" /> is a semi-simple matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438019.png" /> is nilpotent and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438020.png" />; the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438022.png" /> belong to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084380/s08438023.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Algèbre" , ''Eléments de mathématiques'' , '''2''' , Hermann  (1959)</TD></TR></table>