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''of a linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826001.png" /> of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826002.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826003.png" />''
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{{TEX|done}}{{MSC|15A18}}
  
A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826004.png" /> in the kernel of the [[Linear transformation|linear transformation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826007.png" /> is a positive integer depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r0826009.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260010.png" /> is necessarily an eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260011.png" />. If, under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260012.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260013.png" /> is a root vector of height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260014.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260015.png" />.
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''of a linear transformation $A$ of a vector space $V$ over a field $K$''
  
The concept of a root vector generalizes the concept of an [[Eigen vector|eigenvector]] of a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260016.png" />: The eigenvectors are precisely the root vectors of height 1. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260017.png" /> of root vectors belonging to a fixed eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260018.png" /> is a linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260019.png" /> which is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260020.png" />. It is known as the root subspace belonging to the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260021.png" />. Root vectors belonging to different eigenvalues are linearly independent; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260022.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260023.png" />.
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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 $A$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260024.png" /> be finite-dimensional. If all roots of the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260025.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260026.png" /> (e.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260027.png" /> is algebraically closed), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260028.png" /> decomposes into the direct sum of different root spaces:
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The concept of a root vector generalizes the concept of an [[Eigen vector|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 $\lambda \ne \mu$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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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:
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\begin{equation}\label{eq:a1}
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V = V_\alpha \oplus \cdots \oplus V_\delta \ .
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\end{equation}
  
This decomposition is a special case of the weight decomposition of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260030.png" /> relative to a splitting nilpotent Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260031.png" /> of linear transformations: The Lie algebra in this case is the one-dimensional subalgebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260032.png" /> in the Lie algebra of all linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260033.png" /> (see [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260034.png" /> relative to some basis is a [[Jordan matrix|Jordan matrix]], then the components of the decomposition (*) may be described as follows: The root subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260035.png" /> is the linear hull of the set of basis vectors which correspond to Jordan cells with eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082600/r08260036.png" />.
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Voevodin,  "Algèbre linéare" , MIR  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Foundations of linear algebra" , Freeman  (1963)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Voevodin,  "Algèbre linéare" , MIR  (1976)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Foundations of linear algebra" , Freeman  (1963)  (Translated from Russian)</TD></TR>
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</table>

Latest revision as of 19:45, 16 November 2017

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)
How to Cite This Entry:
Root vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Root_vector&oldid=17779
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article