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I'm looking for a class of matrices such that if it contains a matrix with only positive entries then the inverse of said matrix also has only positive entries. I imagine an example of such a class would be the class of orthogonal matrices where the inverse is the transpose but i'm looking for a more general class if possible.

Qtip
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  • https://en.wikipedia.org/wiki/Definite_matrix you are looking for what's called a "positive definite matrix" – Noa Even Sep 08 '21 at 09:51
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    If $A, B$ are "positive" in the sense that all entries are positive, then so does $AB$. This means $B$ cannot be an inverse of $A$. – achille hui Sep 08 '21 at 10:10

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If I understood your question correctly, you can find what you're looking for in "When a Matrix and Its Inverse Are Nonnegative" by J. Ding and N. H. Rhee, where theorem 5.1 states:

"A matrix and its inverse are nonnegative matrices if and only if it is the product of a diagonal matrix with all positive diagonal entries and a permutation matrix."

user23571113
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