Matrix Properties / Features

We often want to describe the set of all matrices of a given size and type.

General Notation: $\mathcal{M}_{n \times m}(S)$ (

$\mathcal{M}_{n \times m}(S)$

) or $\mathcal{M}_{n,m}(S)$ (

$\mathcal{M}_{n,m}(S)$

) denotes the set of all \( n \times m \) matrices whose entries come from the set \( S \). Examples:

• \( \mathcal{M}_{n \times m}(\mathbb{R}) \): all \( n \times m \) matrices with real entries
• \( \mathcal{M}_{n \times m}(\mathbb{Z}) \): all \( n \times m \) matrices with integer entries
• \( \mathcal{M}_{n \times m}(\mathbb{Q}) \): all \( n \times m \) matrices with rational entries
• \( \mathcal{M}_{n \times m}(\mathbb{C}) \): all \( n \times m \) matrices with complex entries

Alternative Notation: $S^{n \times m}$, meaning all \( n \times m \) matrices over the set \( S \).

Example: 1. $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$ so $A \in \mathcal{M}_{2 \times 3}(\mathbb{P})$ or $A \in \mathbb{P}^{2 \times 3}$.