Given a random symmetric matrix (assume each element i.i.d $\sim \mathbf{N}(0, 1)$) $M \in R^{n\times n}$, and it's eigendecomposition $$M = Q \Lambda Q^{-1}$$
What is the maximum element of $Q$?
After running some random tests in python I noticed that the maximal element is always between $0$ and $1$.
What is the reason behind this? I have not been able to come up with an analytical way to describe this phenomena.
This will be true for all orthonormal matrices. Since it is the eigen decomposition of a symmetric matrix, your eigenvectors will form an orthonormal basis. Note that their columns (or rows) are unit-norm. Thus individual entries should have absolute value less than or equal to 1.