Are there any non-discrete definitions for the size of a matrix?

Question

Throughout my math education I have noticed that in order to solve a difficult problem with one set of numbers it helps to move to a larger encompassing set. For example, subtracting some natural numbers, $\mathbb{N}$ , requires the integers, $\mathbb{Z}$ (e.g. $3-4$ would be meaningless in a world with only the natural numbers). This pattern seems to continue, from the integers to the real numbers to the complex numbers. There also seems to be another pattern that holds with the idea of scalars to matrices to tensors, etc. Each next set holding (or generalizing) the previous.

What I have noticed--in my limited math education--is that matrices stick to natural numbers in their dimensionality. That is: $$i,j \in \mathbb{N},\mathbb{R}^{i\times j}$$

My question boils down to this:

1. Can matrix $A \in \mathbb{R}^{i\times j} : i,j \in \mathbb{C}$?
2. If not why not?
3. If so, then what does it mean for one of these objects to have a non-natural dimension? For example, a vector, $v \in \mathbb{R}^{- \pi/2 \times 1} $ or a matrix $A \in \mathbb{R}^{0.5 \times -1} $? Is such an object even able to be represented?

Answer 1

A perhaps related result to what you're after, if not a direct answer:

Given any two sets $P$ and $Q$, you can from the vector spaces $\mathbb{R}^P$ and $\mathbb{R}^Q$ consisting of all functions from $P$ (resp. $Q$) to $\mathbb{R}$. Then the matrices representing linear maps from $\mathbb{R^P} \to \mathbb{R}^Q$ are given by elements of $\mathbb{R}^{P \times Q}$: functions from $P \times Q \to \mathbb{R}$.

These vector spaces have dimension other than natural numbers (say, if $P$ and $Q$ are infinite) but they do not have dimensions like what you might be looking for: i.e. they are not "complex dimensional", etc. Instead, these vector spaces can take cardinal numbers as their dimension.

As far as a "continuous" analogue of dimension goes, I don't know of any reasonable way to make sense of a "vector space of dimension $\frac{1}{2}$", let alone one of dimension $2+i$. It also strikes me as unlikely that such a concept exists. That said, I haven't ever thought to look into such things, and I've been surprised before. Who knows, maybe you'll be the person to develop such a notion!

Edit:

After a short google, it looks like this is something that people have thought of before (at least for rational dimension), but the machinery is rather complex. See here for instance.

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I hope this helps ^_^

Answer 2

What you are proposing is integral kernels, which are essentially matrices with real dimensions, an integral kernel is usually a function of two arguments, each representing a "column" and a "row", but these arguments can be non-integer.

The theory of integral kernels is well developed.

An analog of multiplication of matrix by vector would be

$g(\alpha)=\int_a^b f(t)K(\alpha, t)dt$

An analog of multiplication of matrices is

$r(x,y)=\int_a^b K_1(x,t) K_2(t,y) \, dt$

The miltiplicative unity is played by the Delta destribution: $f(x,y)=\delta(x-y)$, any kernel "multiplied" by this kernel remains the same. It is an analog of Kroneker delta in matrix algebra.

Answer 3

If $A$'s indices can be any complex numbers, $A$ becomes a linear operator that maps a function $f$ of domain $\Bbb C$ to $\int_{\Bbb C^2}A(x, \, y)f(y) dy$. This involves regarding each such function $f$ as a vector in a space of dimension $|\Bbb C|$.