Semi-simple operator
Encyclopedia
In mathematics
, a linear operator T on a finite-dimensional vector space
is semi-simple if every T-invariant subspace
has a complementary T-invariant subspace.
An important result regarding semi-simple operators is that, a linear operator on a finite dimensional vector space over an algebraically closed field is semi-simple if and only if it is diagonalizable. This is because such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane
, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any basis for this space can be extended to an eigenbasis.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a linear operator T on a finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
is semi-simple if every T-invariant subspace
Invariant subspace
In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
has a complementary T-invariant subspace.
An important result regarding semi-simple operators is that, a linear operator on a finite dimensional vector space over an algebraically closed field is semi-simple if and only if it is diagonalizable. This is because such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any basis for this space can be extended to an eigenbasis.