Stochastic Canonical Equation K ₄₃ for Normalized Spectral Functions of Random Gram Block Matrices

Abstract

In this chapter, we consider nonsymmetric block matrices of the form

$$ {\tilde \Xi _{p1\;{\rm{x}}\;p2}} = {\Xi _{p1q1\;{\rm{x }}p2q2}} = \;{\rm{(}}\Xi _{ij}^{(p1,\;{\rm{p2}})}{\rm{)}}_{i = 1,\; \ldots \;{\rm{, }}p{\rm{1}}}^{j = 1,\; \ldots \;{\rm{, }}p{\rm{2}}} $$

with complex matrices $\Xi _{ij}^{({p_1},\;{p_2})}$ of size q ₁ x q ₂. We find stochastic canonical equations for resolvents of corresponding Gram matrices and consider the case where the expectation of random blocks $\Xi _{ij}^{({p_1},\;{p_2})}$ do not exist.