Semi-simple matrix

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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 $F[A]$.


[1] N. Bourbaki, "Algèbre" , Eléments de mathématiques , 2 , Hermann (1959)


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:
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article