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The problem came about to me when I asked:
Do matrices with continuous (real) indices x and y exist?
So, thinking a little further, I though such set of continuous matrices would have to form a field, just like the set of discrete matrices we are familiar with. That would mean we would need to perform arithmetic operation with them, addition, subtraction, multiplication, and division.
I simply defined addition and subtraction as such
f(x,y) + g(x,y)
and
f(x,y) - g(x,y).
Multiplication was a little harder, but still simple:
.
and a multiplicative identity does indeed exist just like discreet matrices. That identity is
.
But the problem I came across is which division (existence of multiplicative inverses). That brings me to the problem I just posted. Solving equations like that is a step in the right direction to show that multiplicative inverses exist, and thus the field of continuous matrices exist.
