# Functional Analysis – Measurement and Characterization of Local Correlation Matrices

Definition: A matrix $$C in mathbb R ^ {m times n}$$ is a local correlation matrix if there are true random variables $$x_1, dots, x_m, y_1, dots, y_n$$ defined on a common probability space in which values ​​are recorded $$[-1,+1]$$ so that $$C_ {ij} = mathbb E[x_iy_j]$$ holds with everyone $$(i, j) in {1, dots, m } times {1, dots, n }$$,

Set of all $$m times n$$ local correlation matrices $$mathsf {LC} _ {m, n}$$,

1. What is the measure of $$mathsf {LC} _ {m, n} cap[-1,+1]^ {m times n}$$ in the $$[-1,+1]^ {m times n}$$?

2. Is there a nontrivial geometric or functional characterization of $$mathsf {LC} _ {m, n}$$?

3. Are there explicit template families in $$[-1,+1]^ {m times n}$$ they are not$$mathsf {LC} _ {m, n}$$?

4. Given a $$m times n$$ Matrix, there is a simple test if it is a member $$mathsf {LC} _ {m, n}$$?