Eigenraum

Eigenspace is a term from linear algebra . It describes the linear envelope of the eigenvectors for a certain eigenvalue of an endomorphism . The eigenvectors thus span a sub-vector space .

A generalization of the eigenspace is the main space . If an eigenvalue has the algebraic multiplicity 1, then eigenspace and main space are the same for this eigenvalue.

definition

Let be a vector space over a body and an endomorphism , that is, a linear map . The eigenspace for the eigenvalue of is then ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle \varphi \in \operatorname {End} (V)}$ ${\displaystyle \varphi \colon V\to V}$ ${\displaystyle E(\lambda )}$ ${\displaystyle \lambda }$ ${\displaystyle \varphi }$

${\displaystyle {\begin{aligned}E(\lambda )&:=\mathrm {Kern} (\varphi -\lambda \mathrm {id} _{V})\\&=\left\{x\in V\mid \varphi (x)=\lambda x\right\}\\&=\left\{x\in V\mid x\neq 0,\ \varphi (x)=\lambda x\right\}\cup \left\{0\right\}\end{aligned}}}$

Here referred to the identity mapping on . ${\displaystyle \mathrm {id} _{V}}$ ${\displaystyle V}$

One then also says is invariant with respect to the endomorphism or is an -invariant subspace of . The elements of are then the eigenvectors of the eigenvalue of , as well as the zero vector . ${\displaystyle E\left(\lambda \right)\subseteq V}$ ${\displaystyle \varphi }$ ${\displaystyle E\left(\lambda \right)}$ ${\displaystyle \varphi }$ ${\displaystyle V}$ ${\displaystyle x}$ ${\displaystyle E\left(\lambda \right)}$ ${\displaystyle \lambda }$ ${\displaystyle \varphi }$

Geometric multiplicity

The dimension of the eigenspace is called the geometric multiplicity of . It is always at least 1 and at most equal to the algebraic multiplicity of . If the dimension of the eigenspace is greater than 1, the eigenvalue is degenerate called, otherwise, it is non-degenerate . ${\displaystyle E\left(\lambda \right)}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle E\left(\lambda \right)}$

properties

• If there is an eigenvalue of , the corresponding eigenspace is equal to the kernel of . For and on the definition of own space: . ${\displaystyle \lambda =0}$ ${\displaystyle \varphi }$ ${\displaystyle E\left(\lambda \right)}$ ${\displaystyle \varphi }$ ${\displaystyle \operatorname {Kern} \left(\varphi \right)=\left\{x\in V\mid \varphi \left(x\right)=0\right\}}$ ${\displaystyle E\left(0\right)=\left\{x\in V\mid \varphi \left(x\right)=0x=0\right\}}$

• The sum of eigenspaces for pairwise different eigenvalues of is direct : ${\displaystyle n}$ ${\displaystyle \lambda _{1},\dotsc ,\lambda _{n}}$ ${\displaystyle \varphi }$

${\displaystyle E(\lambda _{1})+\dots +E(\lambda _{n})=E(\lambda _{1})\oplus \dots \oplus E(\lambda _{n})}$

• If it holds in the above case , then has a basis of eigenvectors of . In this case, each representation matrix of relative to a base of diagonalisierbar, that is the representation matrix of with respect to a base of eigenvectors of has diagonal form . In the main diagonal of are then the eigenvalues ​​of : ${\displaystyle E(\lambda _{1})+\dots +E(\lambda _{n})=V}$ ${\displaystyle V}$ ${\displaystyle \varphi }$ ${\displaystyle A}$ ${\displaystyle \varphi \in \operatorname {End} \left(V\right)}$ ${\displaystyle V}$ ${\displaystyle A'}$ ${\displaystyle \varphi }$ ${\displaystyle V}$ ${\displaystyle \varphi }$ ${\displaystyle A'}$ ${\displaystyle \varphi }$

${\displaystyle A'={\begin{pmatrix}\lambda _{1}&0&\cdots &0\\0&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&\lambda _{n}\end{pmatrix}}}$

• If a Prehilbert space is and is self adjoint , then the eigenspaces of different eigenvalues ​​are orthogonal to each other in pairs . ${\displaystyle V}$ ${\displaystyle \varphi \in \operatorname {End} (V)}$

literature

• Gerd Fischer: Linear Algebra. An introduction for first-year students . 17th updated edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 , ( Studies. Basic course in mathematics ).