A unified matrix model including both CCA and F matrices in multivariate analysis: the largest eigenvalue and its applications

Let $bbZ_{M_1times N}=bbT^{frac{1}{2}}bbX$ where $(bbT^{frac{1}{2}})^2=bbT$ is a positive definite matrix and $bbX$ consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model $$bold{bbom}=(bbZbbU_2bbU_2^TbbZ^T)^{-1}bbZbbU_1bbU_1^TbbZ^T,$$ where $bbU_1$ and $bbU_2$ are isometric with dimensions $Ntimes N_1$ and $Ntimes (N-N_2)$ respectively such that $bbU_1^TbbU_1=bbI_{N_1}$, $bbU_2^TbbU_2=bbI_{N-N_2}$ and $bbU_1^TbbU_2=0$. Moreover, $bbU_1$ and $bbU_2$ (random or non-random) are independent of $bbZ_{M_1times N}$ and with probability tending to one, $rank(bbU_1)=N_1$ and $rank(bbU_2)=N-N_2$. We establish the asymptotic Tracy-Widom distribution for its largest eigenvalue under moment assumptions on $bbX$ when $N_1,N_2$ and $M_1$ are comparable. By selecting appropriate matrices $bbU_1$ and $bbU_2$, the asymptotic distributions of the maximum eigenvalues of the matrices used in Canonical Correlation Analysis (CCA) and of F matrices (including centered and non-center