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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.

1 Answers 1

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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.

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    In addition: The maximum of $1$ can be reached, (for example) when $M$ is already diagonal. The minimum is obtained, when every element of $Q$ has the same magnitude. It is $\sqrt{n}^{-1}$ then. – 2017-02-14
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    Why does the eigendecomposition of a symmetric matrix give a orthonormal basis vs a simply orthogonal basis? – 2017-02-14