Let $C$ be a positive definite correlation matrix partitioned as
$$C=\begin{bmatrix} I_{k_1} & A \\ A' & I_{k_2} \end{bmatrix}$$
How can I show that the eigenvalues of $AA'$ are all less than $1$?
2026-04-02 03:02:51.1775098971
Show the eigenvalues of this submatrix are all less than $1$
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Using the Schur complement of the lower-right entry, we see that $C$ is positive definite if and only if $I_{k_2} - AA'$ is positive definite. This occurs if and only if the eigenvalues of $AA'$ are less than $1$.