Let $H$ be a $n \times n$ real symmetric matrix that has eigenvalues with absolute value less than 1. Define the matrix $M = \prod_{i=1}^n (I - e_ie_i^{\top}H)$ where $e_i$ is the vector with all coordinates zero but the $i$-th, which is 1.

Assuming that $H$ has at least one negative eigenvalue, is it true that $M$ has an eigenvalue with absolute value greater than 1?