As a mathematician I think of matrices as ${\mathbb{F}}^{m\times n}$, where $\mathbb{F}$ is a field and usually $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$. Units are not necessary.

However, some engineers have told me that they prefer matrices to have units such as metres, kg, pounds, dollars, etc. Assigning a unit of measurement to each entry to me seems restrictive (for instance if working with dollars then is ${A}^{2}$ allowed?).

Here are a few things that I would like to understand more deeply:

1. Are there examples where it is more appropriate to work with matrices that have units?

2. If units can only restrict the algebra, why should one assign units at all?

3. Is there anything exciting here, or is it just engineers trying to put physical interpretations on to matrix algebra?

However, some engineers have told me that they prefer matrices to have units such as metres, kg, pounds, dollars, etc. Assigning a unit of measurement to each entry to me seems restrictive (for instance if working with dollars then is ${A}^{2}$ allowed?).

Here are a few things that I would like to understand more deeply:

1. Are there examples where it is more appropriate to work with matrices that have units?

2. If units can only restrict the algebra, why should one assign units at all?

3. Is there anything exciting here, or is it just engineers trying to put physical interpretations on to matrix algebra?