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I'm playing around generating random matrices with either sparse gaussian entries or sparse 0-1 entries.

In both cases, I find that the power method often fails to find an eigenvector (from an all-real seed vector), and that the extended power method (instead of each step being a matrix-vector multiplication and a normalization, each step is a matrix-vector multiplication, an orthogonalization against previous eigenvectors, and a normalization) also tends to fail on the 2nd or 3rd eigenvector.

Are there simple ways to generate an all-real eigenvector?

Or am I wrong about these being complex eigenvalues and I should start looking for a bug in my code?

Elliot JJ
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Let $A$ be a random $n\times n$ matrix s.t. the $(a_{i,j})$ are independent and follow the normal law $N(0,1)$. Let $E_n$ be the mean of the number of real eigenvalues of $A$; then $E_n\sim \sqrt{2/\pi}\sqrt{n}$. Thus the density of real eigenvalues tends to $0$ when $n$ tends to $\infty$. In particular, the eigenvalues that realize $\rho(A)$ are not real with a probability that tends to $1$.