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A totally positive matrix is one whose minors are all positive. This is a simple elementary concept but most of the development on the subject is far from elementary. I am having a hard time understanding most papers on the subject because of the complicated language.

I would like to know, in simple terms, what is known about real totally positive matrices. What are necessary and sufficient conditions for a matrix to be totally positive? What is the simplest known algorithm to verify total positivity?

Thanks in advance.

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up vote 4 down vote accepted

Here's one answer (taken from Fomin and Zelevinsky's review paper, which you can download from here (ref. [2]):

An initial minor is a solid minor (consecutive rows and columns) bordering either the left or the top edge of the matrix. Each matrix entry is the lower right corner of exactly one initial minor, so there are $n^2$ of them (as opposed to $\binom{2n}{n}-1$ if you count all minors). According to Theorem 9 in the paper, a square matrix is totally positive iff all its initial minors are positive, and it can be shown that there is no criterion which can get away with testing less than $n^2$ minors.

And then there are of course many other things known about totally positive matrices; for example, the eigenvalues are positive and simple.

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