قضیه پرون-فربنیوس

${\displaystyle \mathit{x}\mathbf{(}\mathbf{0}\mathbf{)}=\sum _i{c}_i{\mathit{v}}_{\mathit{i}}}$

${\displaystyle \mathit{x}\mathbf{(}\mathit{t}\mathbf{)}={\mathit{A}}^t\mathit{x}\mathbf{(}\mathbf{0}\mathbf{)}={\mathit{A}}^t\sum _i{c}_i{\mathit{v}}_{\mathit{i}}=\sum _i{c}_i{\lambda }_i^t{\mathit{v}}_{\mathit{i}}={\lambda }_1^t\sum _i{c}_i{(\frac{{\lambda }_i}{{\lambda }_1})}^t{\mathit{v}}_{\mathit{i}}}$

${\displaystyle \underset{t\to \mathrm{\infty }}{lim}\frac{\mathit{x}\mathbf{(}\mathit{t}\mathbf{)}}{|{\lambda }_1|}=c_1{\mathit{v}}_{\mathbf{1}}}$