Semi-simple endomorphism
semi-simple linear transformation, of a vector space over a field
An endomorphism of
with the following property: For any
-invariant subspace
of
there exists an
-invariant subspace
such that
is the direct sum of
and
. In other words,
should be a semi-simple module over the ring
,
acting as
. For example, any orthogonal, symmetric or skew-symmetric linear transformation of a finite-dimensional Euclidean space, and also any diagonalizable (i.e. representable by a diagonal matrix with respect to some basis) linear transformation of a finite-dimensional vector space, is a semi-simple endomorphism. The semi-simplicity of an endomorphism is preserved by passage to an invariant subspace
, and to the quotient space
.
Let . An endomorphism
is semi-simple if and only if its minimum polynomial (cf. Matrix) has no multiple factors. Let
be an extension of the field
and let
be the extension of the endomorphism
to the space
. If
is semi-simple, then
is also semi-simple, and if
is separable over
, then the converse is true. An endomorphism
is called absolutely semi-simple if
is semi-simple for any extension
; for this it is necessary and sufficient that the minimum polynomial has no multiple roots in the algebraic closure
of
, that is, that the endomorphism
is diagonalizable.
References
[1] | N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1970) pp. Chapts. I-III |
Comments
Over an algebraically closed field any endomorphism of a finite-dimensional vector space can be decomposed into a sum
of a semi-simple endomorphism
and a nilpotent one
such that
; cf. Jordan decomposition, 2).
Semi-simple endomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_endomorphism&oldid=36910