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We give two characterizations of Perron similarities and study the polyhedra $\\mathcal{C}(S) := \\{ x \\in \\mathbb{R}^n: S D_x S^{-1} \\geq 0,~D_x := \\text{diag}(x) \\}$ and $\\mathcal{P})(S) := \\{x \\in \\mathcal{C}(S) : x_1 = 1 \\}$, which we call the \\emph{Perron spectracone} and \\emph{Perron spectratope}, respectively. 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Johnson, Pietro Paparella","submitted_at":"2015-08-29T05:27:45Z","abstract_excerpt":"Call an $n$-by-$n$ invertible matrix $S$ a \\emph{Perron similarity} if there is a real non-scalar diagonal matrix $D$ such that $S D S^{-1}$ is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra $\\mathcal{C}(S) := \\{ x \\in \\mathbb{R}^n: S D_x S^{-1} \\geq 0,~D_x := \\text{diag}(x) \\}$ and $\\mathcal{P})(S) := \\{x \\in \\mathcal{C}(S) : x_1 = 1 \\}$, which we call the \\emph{Perron spectracone} and \\emph{Perron spectratope}, respectively. 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