estuff

Eigenvalues and Eigenvectors

Definitions

An eigenvector

V

associated with a matrix

M

is a vector whose image under

M

is a simple scaling, thus:

M \otimes V = λ V

Note that every eigenvalue is thus associated with a 1-dimensional linear subspace defined by it\'s eigen vector, since

\forall k \in ℝ

M \otimes \vec{V} = λ \vec{V} \Rightarrow \vec{V} \otimes (k V) = k λ \vec{V}

Examples

Another way to think of the line defined by the eigenvector is as an invariant subspace - every 'point' along that line is mapped to that line by the action of

M

.
In dynamical systems, the 'acceleration' or 'flow' is often represented as a field of vectors, each assigned to a position

\vec{X} = (x_{1}, y_{1})

by the equation

\vec{X} = M \otimes \vec{X}

, usually approximated as

\vec{X_{i + 1}} = \vec{X_{i}} + M \otimes \vec{X}

. This means the dynamics along these lines is known - and by 'continuity', they must be 'similar nearby'. Thus, understanding the eigenstuff yields understanding about the overall behavior of the system:

M = [\begin{matrix} - 2 & 1 \\ 1 & - 2 \end{matrix}]

has eigenvectors

\vec{V_{1}} = (1,1)

and

\vec{V_{2}} = (1, - 1)

, with eigenvalues

λ_{1,2} = - 1

(why?). The phase portrait is 'fixed' by the dynamics of these subspaces: