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I'm stuck to prove the following exercise : Given real numbers $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$, show that $$ \det(e^{\large{x_iy_j}})_{i,j=1}^n>0 $$ provided that $x_1<\cdots and $y_1<\cdots.

Any idea ?

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    I assume that the index of $y$ is $j$.2012-05-26
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    @Phira : You're right. Let me edit that.2012-05-26
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    If one of the sets are integers, this follows from the Vandermonde determinant and the definition of a Schur function.2012-05-26
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    Can you give some context for the exercice?2012-05-26
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    In approximation theory, this is about to show that the family of functions $x\mapsto e^{xy_i}$, $i=1,\ldots,n$, forms a Chebyshev system.2012-05-26
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    Positivity of this matrix is equivalent to positivity of another matrix $(e^(x_i-x_j)^2)$ (expand the square then one gets $x_iy_j$). But the latter is well known to be positive, for example in the area of gaussian processes (it is the covariance matrix). There is actually a very neat lower bound for the determinant of the matrix you give, proved by Drury-Marshall (1987).2012-05-30
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    This has been proved in Theorem 6.5.2, pp.297-299 of Kung *et al.*, [Combinatorics: The Rota Way](http://www.math.tamu.edu/~cyan/book.html), Cambridge University Press.2013-05-09

3 Answers 3