Root vector
of a linear transformation of a vector space
over a field
A vector in the kernel of the linear transformation
, where
and
is a positive integer depending on
and
. The number
is necessarily an eigenvalue of
. If, under these conditions,
, one says that
is a root vector of height
belonging to
.
The concept of a root vector generalizes the concept of an eigenvector of a transformation : The eigenvectors are precisely the root vectors of height 1. The set
of root vectors belonging to a fixed eigenvalue
is a linear subspace of
which is invariant under
. It is known as the root subspace belonging to the eigenvalue
. Root vectors belonging to different eigenvalues are linearly independent; in particular,
if
.
Let be finite-dimensional. If all roots of the characteristic polynomial of
are in
(e.g. if
is algebraically closed), then
decomposes into the direct sum of different root spaces:
![]() | (*) |
This decomposition is a special case of the weight decomposition of a vector space relative to a splitting nilpotent Lie algebra
of linear transformations: The Lie algebra in this case is the one-dimensional subalgebra generated by
in the Lie algebra of all linear transformations of
(see Weight of a representation of a Lie algebra).
If the matrix of relative to some basis is a Jordan matrix, then the components of the decomposition (*) may be described as follows: The root subspace
is the linear hull of the set of basis vectors which correspond to Jordan cells with eigenvalue
.
References
[1] | V.V. Voevodin, "Algèbre linéare" , MIR (1976) (Translated from Russian) |
[2] | A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) |
Root vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Root_vector&oldid=42304