The result of $a+0(b)$, where a and b are matrices of different dimensions.

Question

Assume we have matrices $a$ and $b$ that have different dimensions.

Then, is the expression $a$+$0(b)$ equal to $a$ or undefined?

Answer 1

The short answer is that it is undefined. But can you define it? Sure. You can add together any two things you want. The question then becomes: what does this mean?

I'm going to give you an example of an intelligent way to do this. When we treat the elements of $\mathbb R^2$ as complex numbers, we say that $\mathbb R$ is a subset of $\mathbb R^2$, namely the $x$-axis, $\mathbb R\times\{0\}$. In the same way we can identify $\mathbb{R}^{n-1}$ with the subset $\mathbb R^{n-1}\times\{0\}$ of $\mathbb R^n$. Thus we can consider $\mathbb R^m$ as a particular fixed subset of $\mathbb R^n$ whenever $m\leq n$. We denote the union $$\bigcup_{n=1}^\infty\mathbb R^n$$ by $\mathbb R^\infty$.

An alternative, convenient way to think about this set is as the set of all infinite sequences of real numbers where all but finitely many terms are $0$. Then the set of all sequences where all but the first $n$ terms are $0$ is in fact $\mathbb R^n$.

Now, if $a$ is an $m\times n$ matrix, then $a$ is a function from $\mathbb R^n$ to $\mathbb R^m$, and hence can have its codomain extended so that it is a function $\mathbb R^n\to \mathbb R^\infty$. We can also extend the domain to all of $\mathbb R^\infty$ by declaring that $a$ just ignores all terms but the first $n$. Then $a$ is a linear function $\mathbb R^\infty\to\mathbb R^\infty$.

Now no one can argue that in this situation the sum $a+0b$ is defined and equal to $a$ for any two finite matrices $a$ and $b$.

Nobody has fooled you, though, by saying you can't add matrices of different sizes together, because this is a rather contrived situation that I just made up. I don't know why you would ever want to do this, but maybe it's useful for something.