Eigenvalue and Eigenvector

Defn

A vector \(v\in V\) is a eigenvector if \(v\neq 0\) and \(\exists \lambda\in F\) st \(T(v)=\lambda v\). The \(\lambda\) is te eigenvalue

\(T:V\rightarrow V\) is diagonalizable iff \(V\) has a basis \(\beta\) that are all eigenvectors

Furthermore, \([T]_\beta=D\) is a diagonal matrix and \(d_{jj}\) is the eigenvalue corresponding to \(\beta_j\)

\(\lambda\in F\) is an eigenvalue of \(A\iff det(A-\lambda I)=0\)

\(\lambda\) is an eigenvalue of \(A\iff\exists v\in V\) none zero st \(Av=\lambda v\)

\(\iff (A-\lambda I)v=0\)

\(\iff (A-\lambda I)x=0\) has a non-zero solution

\(\iff (A-\lambda I)\) is not invertible

A characteristic polynomial of square matrix \(A\) is \(det(A-t I)\)

A vector \(v\in V\) is an eigenvector of \(T\) corresponding to \(\lambda\) iff \(v\neq0\) and \(v\in N(T-\lambda I)\)