Let \(T: V\rightarrow V\) be an endomorphism (maps from a space to itself) of a vector space \(V\). A vector \(v_0 \ne 0\) of \(V\) is an eigenvector of \(T\) with respect to the eigenvalue \(\lambda\) if:

\[T(v_0)=\lambda v_0\]

The set of eigenvalues is called the spectre of \(T\). If \(\lambda \in \ space(T)\) then the set:

\[V_{\lambda} = \{ v\in T\ |\ T(v)=\lambda v \}\]

is called eigenspace.

Notice that from the definition follows:

\[Tv_0 - \lambda v_0 = 0\] \[(T - \lambda I)v_0 = 0\]

Therefore \((T-\lambda I)=0\) (is singular) and It's determinant is 0.

\[det(T-\lambda I)=0\]

This is very useful to compute the eigenvalues. Once you have those, and T, you can use them in the definition to get the eigenspace and the eigenvectors.

---

Travel: Mathematics Notes, Index