Let $\mathcal{S}_n$ denote the space of symmetric positive definite matrices. Suppose $\mathbf{R} \in \mathcal{S}_n$ has smallest eigenvalue $\lambda_{min} > 1/2$ and is entry-wise positive. Given some scalar function $f$, define the corresponding spectral function $F(\mathbf{R}):\mathcal{S}_n \to \mathcal{S}_n$ such that $F(\mathbf{R})$ shares its eigenvectors with $\mathbf{R}$ yet the eigenvalues are replaced by $f(\lambda_i)$, respectively. Fix some $n$-dimensional non-negative vector $\mathbf{x}$.

For which spectral functions $F$ do we have $F(\mathbf{R}) \mathbf{x} \geq 0$?

It is known that non-negativity of matrices of general dimension is preserved iff $f$ is absolutely monotonic. While sufficient, I wonder whether a weaker condition will do. Something that may matter is that the angle between a vector and its image under a positive-definite matrix is bounded by 90°.

I further wonder whether additional assumptions on $\mathbf{R}$ may help, e.g. strict diagonal dominance.



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