# Probability of increase in $\frac{\lambda}{n}$ of correlation matrix (principal score) after random sample the original matrix

Suppose $X$ is the original matrix of $n$ columns, and $P$ is the $n \times n$ correlation matrix of $X$. $P$ is symmetric positive semi-definite.

Denote the largest eigenvalue of $P$ is $\lambda$, then we must have $\frac{\lambda}{n} \in [0,1]$ and it is called the principal score.

Randomly sample $k$ columns $(k <n)$ from $X$ and compute the correlation matrix $P_0$ of this sampled subset. Denote the new principal score as $\frac{\lambda_0}{k}$. The problem is

If $m$ columns in $X$ has all their mutual correlations greater than $\frac{\lambda}{n}$, is there any way to estimate probability that $\frac{\lambda_0}{k} \ge \frac{\lambda}{n}$?