Ridicarea la putere a unei matrice

$Definiție. Fie A\in {\mathcal{M}}_n\left(\mathrm{ℂ}\right). A^0=I_n, A^1=A, A^2= A·A, A^n=\underset{n ori}{\underbrace{A·A·...A}}, n\in {\mathrm{\mathbb{N}}}^{*}.$

Reguli de calcul

$1. A^k·A^p=A^{k+p}, \forall A,B\in {\mathcal{M}}_n\left(\mathrm{ℂ}\right), k,p\in \mathrm{\mathbb{N}}$

$2. (A^k{)}^p=A^{kp}, \forall A\in {\mathcal{M}}_n(\mathrm{ℂ}), k,p\in \mathrm{\mathbb{N}}.$

$3. (A+B{)}^2=A^2+AB+BA+B^2, \forall A,B\in {\mathcal{M}}_n(C).$

$3^{`}. Dacă n\ge 2, n\in \mathrm{\mathbb{N}}, A,B\in {\mathcal{M}}_n\left(\mathrm{ℂ}\right), AB=BA, (A+B{)}^2=A^2+2AB+B^2.$

$3^{``}. Dacă n,p\ge 2, n,p\in \mathrm{\mathbb{N}}, A,B\in {\mathcal{M}}_n\left(\mathrm{ℂ}\right), AB=BA, (A+B{)}^p=\underset{k=0}{\overset{p}{\sum }}{C}_p^k{A}^{p-k}{B}^k.$

(Formula binomului lui Newton pentru matrice)

$4. Dacă m,n,p\ge 2, m,n,p\in \mathrm{\mathbb{N}}, A,B\in {\mathcal{M}}_n\left(\mathrm{ℂ}\right), AB=BA, A^m{B}^p=B^p{A}^m.$

$Dacă A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathcal{M}}_2\left(\mathrm{ℂ}\right), atunci A^2-(a+d)A+(ad-bc)I_2=O_2. (Relația lui Hamilton- Caylay)$