Mathematics-Online course: Linear Algebra-Normal Forms-Eigenvalues and Eigenvectors-Eigenvalue and Eigenvector

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	Mathematics-Online course: Linear Algebra - Normal Forms - Eigenvalues and Eigenvectors
	Eigenvalue and Eigenvector

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A scalar $\lambda$ is called eigenvalue of square matrix

$\displaystyle A v = \lambda v,\quad v\ne 0\,.$

The vectors $v \neq 0$ with $Av = \lambda v$ are called eigenvectors associated with eigenvalue $\lambda$ . The set of the zero vector and of all eigenvectors associated with a given eigenvalue forms a vector space called eigenspace $V_\lambda$ of $\lambda$ .

(Authors: Burkhardt/Höllig/Hörner/Kimmerle)

The matrix

$\displaystyle \left( \begin{array}{rrr}1 & 2 & -1 \\ 0 & 2 & 0 \\ -1 & 2 & 1 \end{array} \right)$

has the eigenvalues $\lambda_1=0$ and $\lambda_2=2$ , since

$\displaystyle \left( \begin{array}{rrr}1 & 2 & -1 \\ 0 & 2 & 0 \\ -1 & 2 & 1 \... ...\end{array} \right)=\left( \begin{array}{r}0 \\ 0 \\ 0 \end{array} \right)$

and

$\displaystyle \left( \begin{array}{rrr}1 & 2 & -1 \\ 0 & 2 & 0 \\ -1 & 2 & 1 \... ...rray} \right)=2 \left( \begin{array}{r}1 \\ 1 \\ 1 \end{array} \right)\,.$

(Authors: Burkhardt/Höllig/Hörner)

The following examples illustrate the possible cases for real $2\times2$ -matrices.

two real eigenvalues:

$\displaystyle A=\left(\begin{array}{rr}1& 2\\ 2 & 1\end{array}\right)\,, \quad... ... \quad \lambda_2=3\,,\,v_2=\left(\begin{array}{r} 1\\ 1\end{array}\right)\,.$
one real eigenvalue, two linearly independent eigenvectors:

$\displaystyle A=\left(\begin{array}{rr}2& 0\\ 0 & 2\end{array}\right)\,, \quad... ...end{array}\right)\,, \,v_2=\left(\begin{array}{r} 0\\ 1\end{array}\right)\,.$
one eigenvalue, one eigenvector:

$\displaystyle A=\left(\begin{array}{rr}1& -1\\ 0 & 1\end{array}\right)\,, \quad \lambda=1\,,\,v_1=\left(\begin{array}{r}1\\ 0\end{array}\right)\,.$
no real eigenvalue (two complex eigenvalues):

$\displaystyle A=\left(\begin{array}{rr}1& -1\\ 1 & 1\end{array}\right)\,, \quad \lambda_{1,2}=1\pm \mathrm{i}\,.$

(Authors: Burkhardt/Höllig/Hörner)

The following examples illustrate the possible cases for complex $2\times2$ -matrices.

two eigenvalues:

$\displaystyle A=\left(\begin{array}{rr}\mathrm{i}& 2\\ 2 & \mathrm{i}\end{array... ...bda_2=-2+\mathrm{i}\,,\,v_2=\left(\begin{array}{r}1\\ -1\end{array}\right)\,.$
one eigenvalue, two linearly independent eigenvectors:

$\displaystyle A=\left(\begin{array}{rr}2\mathrm{i}& 0\\ 0 & 2\mathrm{i}\end{arr... ...end{array}\right)\,, \,v_2=\left(\begin{array}{r} 0\\ 1\end{array}\right)\,.$
one eigenvalue, one eigenvector:

$\displaystyle A=\left(\begin{array}{rr}\mathrm{i}& -1\\ 0 & \mathrm{i}\end{arra... ...d \lambda=\mathrm{i}\,,\,v_1=\left(\begin{array}{r}1\\ 0\end{array}\right)\,.$

(Authors: Burkhardt/Höllig/Hörner)

automatically generated 4/21/2005