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A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries) is totally positive. It seems that this fact should be classic. Although I can prove it by a variational argument, I cannot find a reference (in books I can think of or on the Internet) and I would like to know whether this is the "standard" way of proving the result, or if there is another (algebraic?) method known to the community.

Any pointer to a reference or direct proof would be very much appreciated !

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    $\begingroup$ (+1) For this interesting question. May I ask you to include the sketch of your variational proof, in your question. Thank you. $\endgroup$ – Ali Taghavi Nov 1 '17 at 3:58
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    $\begingroup$ By induction. You order the entries like this: $x_1<x_2<\cdots<x_n$, then introduce the variable $x_1=:h$ and set $x_j=:\hat{x}_j+h$ for all $j>1$. By differentiating enough time with respect to $h$ you end up with a Vandermonde of size smaller by $1$, which is positive by hypothesis. $\endgroup$ – Loïc Teyssier Nov 2 '17 at 6:19
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    $\begingroup$ @AliTaghavi: actually, this proof is not mine but is due to A. Glutsyuk, who really was the person looking for references :) $\endgroup$ – Loïc Teyssier Nov 2 '17 at 11:51
  • $\begingroup$ But just a question. Are you considering a particular definition of "positivity"? The standard one required the symetricity of our matrix. This is not the case for the vandermond matrix $\endgroup$ – Ali Taghavi Nov 2 '17 at 22:50
  • $\begingroup$ I gave a definition at the beginning of the question. It is not related to positive definiteness (see Wikipedia). $\endgroup$ – Loïc Teyssier Nov 3 '17 at 6:37
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Totally Non-Negative Matrices Johnson and Fallat, Princeton University Press. Chapters 0 and 1.

Allan Pinkus (2009), Totally Positive Matrices, Cambridge University Press, ISBN 9780521194082

Perhaps it's also in works of Karlin and McGregor? I am looking...

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Total positivity: tests and parametrizations by Fomin and Zelevinsky

According to this reference (bottom of page 5), Fekete proved that a sufficient condition for total positivity is that all solid minors have positive determinant. If this is applied to a Vandermonde matrix $V = (x_i^{j-1})_{ij}$, then positivity of solid minors follows from the formula $$\det V = \prod_{1\leq i < j\leq n} (x_j-x_i) >0,$$ up to factoring out the appropriate (positive) scaling of each row.

The first section of this reference goes through a proof of the stronger result that it suffices to check positivity of all initial solid minors, meaning solid minors that lie along the top row or along the left-most column. They attribute this result to Gasca and Peña, and independently to Cryer.

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The minors of a Vandermonde matrix are known as alternants. Their positivity follows from the fact that they are products of the Vandermonde determinant of the variables involved with a Schur function in these variables. For a short and self-contained proof of this fact, see "Corollary (The Bi-Alternant Formula)" in John R. Stembridge, A Concise Proof of the Littlewood-Richardson Rule, The Electronic Journal of Combinatorics 9 (2002), Note #N5. This note itself is the distillate of several years of algebraic combinatorics (ideas of Lindstrom, Gessel, Viennot, Gasharov, Bender and Knuth are all in there), and longer proofs have been found before (e.g., in Macdonald's book).

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V. Prasolov's book Problems and Theorems in Linear Algebra contains it as Theorem 1.2.12.2. Russian text in pdf Alas, this very useful book seems to be not translated into English. The previous, translated version (AMS 1994) does not contain this result.

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