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3 added non-normal condition

Suppose $A$ is a non-symmetric matrix (also, not a normal matrix) with all non-negative eigenvalues. Is there a relation between eigenspace (subspace spanned by eigenvectors) of $A$ and eigenspace of $(A+A^T)$? Is there an overlap? One obvious observation is that row space of $A$ is same as column space of $A^T$.

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# eigenspace of a non-symmetric matrix and its transpose

Suppose $A$ is a non-symmetric matrix with all non-negative eigenvalues. Is there a relation between eigenspace (subspace spanned by eigenvectors) of $A$ and eigenspace of $(A+A^T)$? Is there an overlap? One obvious observation is that row space of $A$ is same as column space of $A^T$.