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| − | 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]]). | + | A square [[matrix]] over a field $F$ [[Similar matrices|similar]] to a matrix in block diagonal form $\mathrm{diag}[D_1,\ldots,D_k]$, where each $D_i$ is a matrix over $F$ whose [[characteristic polynomial]] is irreducible in $F[X]$, $j=1,\ldots,l$ (cf. [[Irreducible polynomial]]). For a matrix $A$ over a field $F$, the following three statements are equivalent: 1) $A$ is semi-simple; 2) the [[Minimal polynomial of a matrix|minimal polynomial]] of $A$ has no multiple factors in $F[X]$; and 3) the algebra $F[A]$ is a [[semi-simple algebra]]. |
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| − | 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" />. | + | If $F$ is a [[perfect field]], then a semi-simple matrix over $F$ is similar to a diagonal matrix over a certain extension of $F$. For any square matrix $A$ over a perfect field there is a unique representation in the form $A = A_S + A_N$, where $A_S$ is a semi-simple matrix, $A_N$ is nilpotent and $A_SA_N = A_NA_S$; the matrices $A_S$ and $A_N$ belong to the algebra $F[A]$. |
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| | ====References==== | | ====References==== |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre" , ''Eléments de mathématiques'' , '''2''' , Hermann (1959)</TD></TR></table> | + | <table> |
| | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre" , ''Eléments de mathématiques'' , '''2''' , Hermann (1959)</TD></TR> |
| | + | </table> |
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| | + | {{TEX|done}} |
Revision as of 18:03, 12 November 2017
A square matrix over a field $F$ similar to a matrix in block diagonal form $\mathrm{diag}[D_1,\ldots,D_k]$, where each $D_i$ is a matrix over $F$ whose characteristic polynomial is irreducible in $F[X]$, $j=1,\ldots,l$ (cf. Irreducible polynomial). For a matrix $A$ over a field $F$, the following three statements are equivalent: 1) $A$ is semi-simple; 2) the minimal polynomial of $A$ has no multiple factors in $F[X]$; and 3) the algebra $F[A]$ is a semi-simple algebra.
If $F$ is a perfect field, then a semi-simple matrix over $F$ is similar to a diagonal matrix over a certain extension of $F$. For any square matrix $A$ over a perfect field there is a unique representation in the form $A = A_S + A_N$, where $A_S$ is a semi-simple matrix, $A_N$ is nilpotent and $A_SA_N = A_NA_S$; the matrices $A_S$ and $A_N$ belong to the algebra $F[A]$.
References
| [1] | N. Bourbaki, "Algèbre" , Eléments de mathématiques , 2 , Hermann (1959) |
How to Cite This Entry:
Semi-simple matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_matrix&oldid=12282
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article