Powers of a Square Matrix

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$, let $M,B\in\mathcal{M}_{n}(\mathbb{R})$, let $c\in\mathbb{R}$ and let $j,k\in\mathbb{N}_{0}$. The matrix product is a binary operation on $\mathcal{M}_{n}(\mathbb{R})$ with neutral element $I_{n}$, by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv); $M^{k}$ is the power of $M$ for this operation, as in Powers with Exponents in the Natural Numbers with Zero (i) and Powers with Exponents in the Natural Numbers with Zero (ii). Powers $c^{k}$ and $\lVert M\rVert^{k}$ in $\mathbb{R}$ are those of Powers with Exponents in the Natural Numbers with Zero for the multiplication of $\mathbb{R}$.

(i) $M^{0}=I_{n}$, and $M^{k+1}=M^{k}M=MM^{k}$.

(ii) $M^{j+k}=M^{j}M^{k}$.

(iii) If $BM=MB$, then $BM^{k}=M^{k}B$.

(iv) $(cM)^{k}=c^{k}M^{k}$.

(v) $(M^{\top})^{k}=(M^{k})^{\top}$.

(vi) $\lVert M^{k}\rVert\le\lVert M\rVert^{k}$.