Root vector

2020 Mathematics Subject Classification: Primary: 15A18 [MSN][ZBL]

of a linear transformation $A$ of a vector space $V$ over a field $K$

A vector $v$ in the kernel of the linear transformation $(A-\lambda I)^n$, where $\lambda \in K$ and $n$ is a positive integer depending on $A$ and $v$. The number $\lambda$ is necessarily an eigenvalue of $A$. If, under these conditions, $(A - \lambda I)^{n-1}v \ne 0$, one says that $v$ is a root vector of height $n$ belonging to $A$.

The concept of a root vector generalizes the concept of an eigenvector of a transformation $A$: The eigenvectors are precisely the root vectors of height $1$. The set $V_\lambda$ of root vectors belonging to a fixed eigenvalue $\lambda$ is a linear subspace of $V$ which is invariant under $A$. It is known as the root subspace belonging to the eigenvalue $\lambda$. Root vectors belonging to different eigenvalues are linearly independent; in particular, $V_\lambda \cap V_\mu = 0$ if $\lambda \ne \mu$.

Let $V$ be finite-dimensional. If all roots of the characteristic polynomial of $A$ are in $K$ (e.g. if $K$ is algebraically closed), then $V$ decomposes into the direct sum of different root spaces: $$\begin{equation}\label{eq:a1} V = V_\alpha \oplus \cdots \oplus V_\delta \ . \end{equation}$$

This decomposition is a special case of the weight decomposition of a vector space $V$ relative to a splitting nilpotent Lie algebra $L$ of linear transformations: The Lie algebra in this case is the one-dimensional subalgebra generated by $A$ in the Lie algebra of all linear transformations of $V$ (see Weight of a representation of a Lie algebra).

If the matrix of $A$ relative to some basis is a Jordan matrix, then the components of the decomposition $\eqref{eq:a1}$ may be described as follows: The root subspace $V_\lambda$ is the linear hull of the set of basis vectors which correspond to Jordan cells with eigenvalue $\lambda$.

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)