# Difference between revisions of "Semi-simple matrix"

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− | 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, | + | 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 $F[X]$, $j=1,\ldots,k$ (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]]. |

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

</table> | </table> | ||

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+ | ====Comment==== | ||

+ | A [[semi-simple endomorphism]] $\alpha$ of a finite-dimensional vector space $V$ over a field is one for which the matrix of $\alpha$ with respect to some, and hence every, basis of $V$ is semi-simple. | ||

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## Latest revision as of 18:07, 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 $F[X]$, $j=1,\ldots,k$ (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) |

#### Comment

A semi-simple endomorphism $\alpha$ of a finite-dimensional vector space $V$ over a field is one for which the matrix of $\alpha$ with respect to some, and hence every, basis of $V$ is semi-simple.

**How to Cite This Entry:**

Semi-simple matrix.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_matrix&oldid=42278