# Matrix Addition over different fields

We're defining Matrix addition as $$M(k,n) \times M(k,n) \rightarrow M(k,n)$$. Can we add two matrices, A = $$M(k,n)$$ and $$B = M(k,n)$$ of the same size but over different fields? For example, A is a vector space over $$\mathbb{R}$$ and B over $$\mathbb{C}$$? Does this make sense?

• Can you make sense of addition in two different fields? Because that's what you want to do, just in $kn$ independent slots. – Randall Nov 8 '19 at 17:06
• Also, your example question isn't the best since $\mathbb{R}$ is naturally a subfield of $\mathbb{C}$, so really you're just adding in $\mathbb{C}$. You should ponder your question in something like $\mathbb{Z}_2$ and $\mathbb{R}$. – Randall Nov 8 '19 at 17:07
• @Randall So how can I change my example where one is not a sub-field of the other? I noticed that you can add elements in $\mathbb{R}$ to elements in $\mathbb{C}$ because you are just adding real numbers and not changing the complex part. – NinetyNines Nov 8 '19 at 17:10
• Well in my view you cannot do this. This violates all sorts of operational philosophies in algebra. It just doesn't make sense. – Randall Nov 8 '19 at 17:12
• The part where a field is not a sub-field of another field, or addition in $\mathbb{C}$? – NinetyNines Nov 8 '19 at 17:16