relationship between multiplication and correlation

Question

is there a deep interpretation of multiplication as correlation? is this in some sense the most fundamental way to "combine" objects (eg numbers) into products? my reasons for asking are that the correlation function is just multiplication, and matrix multiplication just finds the projection of vectors onto each other, which is just correlation. other examples?

does this all stem from the simple rule that the product of like-signs is positive and the product of opposite-signs is negative? what is the most general statement here? how do other kinds of products fit in?

Answer 1

One can interpret random variables as vectors in an inner product vector space. The expected value of the product of random variables is the inner product function. With this interpretation, correlation is just a vector projection.

Answer 2

Hint: well the two notions some time are so confusing. But another way to make the different is to see correlation as a convolution with a slit different on the impulse phase.