Real Matrix and the Set of Real Matrices

Let $m$ and $n$ be natural numbers, and let $\mathbb{R}$ be the set of real numbers. Write $[m]$ and $[n]$ for the initial segments determined by $m$ and by $n$.

A real $m\times n$ matrix is a function $A$ from the Cartesian product $[m]\times[n]$ to $\mathbb{R}$. For $i\in[m]$ and $j\in[n]$ we write $A_{ij}$ for the value of $A$ at $(i,j)$ and call it the entry of $A$ in row $i$ and column $j$; two real $m\times n$ matrices are equal exactly when all their entries agree.

The set of all real $m\times n$ matrices is denoted $\mathcal{M}_{m\times n}(\mathbb{R})$. A real $n\times n$ matrix is called square, and $\mathcal{M}_{n}(\mathbb{R})$ abbreviates $\mathcal{M}_{n\times n}(\mathbb{R})$.