Notation for the set of m by n matrices

Notation for the set of $m \times n$ matrices: The set of all $m \times n$ matrices with entries from a field $\mathbb{F}$ is denoted by:

$$M_{m \times n}(\mathbb{F}) \quad \text{or} \quad \mathbb{F}^{m \times n} \quad \text{or} \quad \text{Mat}_{m \times n}(\mathbb{F})$$

Common special cases:

• $M_{m \times n}(\mathbb{R})$ or $\mathbb{R}^{m \times n}$: real $m \times n$ matrices
• $M_{m \times n}(\mathbb{C})$ or $\mathbb{C}^{m \times n}$: complex $m \times n$ matrices

When $m = n$, we write $M_n(\mathbb{F})$ or $\mathbb{F}^{n \times n}$ for the set of $n \times n$ square matrices.

This set forms a vector space over $\mathbb{F}$ of dimension $mn$ under matrix addition and scalar multiplication.