Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not orthogonal. Consider $P_i = I-a_ia_i^{'}$, which is an orthogonal projection to the orthogonal complement of $a_i$. I want to find the norm (or largest eigenvalue since they are the same) of the symmetric matrix $$P = P_1...P_{n-1}P_nP_{n-1}....P_1$$

Obviously the norm of $P$ is less than or equal to 1, since all $P_i$ has norm 1. If A is an orthonormal matrix, then $P=0$ since every $P_i$ eliminates a indepedent direction in $\mathbb{R}^n$. But for general case, where A is an arbitary matrix with unit column norm, things are unclear.

If any of the $P_i$ are removed from the product to form $\hat{P}$ (for example, $\hat{P}$ is obtained by removing the $P_1$ on left and right from the product), then the norm of $P'$ is exactly 1. This makes sense because in this case, we can find a vector that is in orthogonal complement of $span\{a_2,...,a_n\}$ and it is easy to check this vector is an eigenvector of $\hat{P}$ with eigenvalue 1.

Some observations are obtained from numerical experiments. If I let $A$ to be randomly generated PD matrix whose columns has norm 1, then P has rank n-1.And the norm of $P$ is usually very close to 1 (>0.99 in all trys). However, these results are certainly not general, considering matrix A with orthonormal columns.

Any thoughts that can explain the result of the numerical experiments? And any idea on finding the norm of $P$ given a particular A? Intuitively it has something to do with the norm of A, but I am not sure. Thanks!