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} $?