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In this paper, we consider a data matrix $X_N\in\mathbb{R}^{N\times p}$ whereall the rows are i.i.d. samples in $\mathbb{R}^p$ of mean zero and covariancematrix $\Sigma\in\mathbb{R}^{p\times p}$. Here the population matrix $\Sigma$is of finite rank perturbation of the identity matrix. This is the "spikedpopulation model" first proposed by Johnstone. As $N, p\to\infty$ but $N/p \to\gamma\in(1, \infty)$, for the sample covariance matrix $S_N := X_NX_N^T/N$, weestablish the joint distribution of the largest and the smallest few packs ofeigenvalues. Inside each pack, they will behave the same as the eigenvaluesdrawn from a Gaussian matrix of the corresponding size. Among different packs,we also calculate the covariance between the Gaussian matrices entries. As acorollary, if all the rows of the data matrix are Gaussian, then these packswill be asymptotically independent. Also, the asymptotic behavior of sampleeigenvectors are obtained. Their local fluctuation is also Gaussian withcovariance explicitly calculated.

Asymptotic Joint Distribution of Extreme Sample Eigenvalues and Eigenvectors in the Spiked Population Model