Continuous matrix multiplication

I thought of this, consider two functions in x,y like for example these:

The first matrix is defined over an x range of -10..10 and the second over a y range of -10..10, so I figured as long as these match then with this formula:

We can do a continuous analogue of matrix multiplication. The height of each function at every point is the value of each matrix in each of it's infinite cells. Then the formula given sums the product of each item in the row in the first matrix with each element in the column in the second matrix just as you would with regular matrix multiplication.

The result:

I'm sure that there is an identity function for this operation, but I'm not sure if or what for example a function that rotates the original function after multiplication would look like...