On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices

Abstract

In this paper, we introduce and analyze a new operation $\circ_{R}$ which mixes two distributions $Ω$ and $Ω'$ via a random orthogonal matrix. In particular, we take $Ω\circ_R Ω'$ to be the limit as $n \to \infty$ of the distribution of singular values of $DRD'$ where $D$ and $D'$ are $n \times n$ diagonal matrices whose diagonal entries have distributions $Ω$ and $Ω'$ respectively and $R$ is a random $n \times n$ orthogonal matrix. We show that $\circ_R$ has several nice properties. We first observe that $\circ_R$ is commutative and associative and compute the moments of $Ω\circ_R Ω'$ in terms of the moments of $Ω$ and $Ω'$. We then show that $\circ_R$ interacts very nicely with the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices. This allows us to answer the question posed by our previous paper of how to describe the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices when the input distribution is not $\{-1,1\}$. In our analysis, we show that the moments of our distributions are closely connected to non-crossing partitions and prove a number of new results on non-crossing partitions which may be of independent interest.