Mathematik-Online lexicon: Real Eigenvalues of 2x2-Matrices

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	Mathematik-Online lexicon:
	Real Eigenvalues of 2x2-Matrices

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

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):

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

see also:

automatisch erstellt am 25. 6. 2018