Given a strictly increasing sequence $0<x\_1<x\_2<\dots<x\_n$ of $n$ strictly positive real numbers and a second strictly increasing sequence $e\_1<\dots e\_n$
of $n$ real numbers, the matrix with coefficients $x\_i^{e\_j}$ has experimentally always $n$ strictly positive real eigenvalues. This matrix is of course a 
Vandermonde matrix if $e\_i=i-1$.


Are there counterexamples to this observation?


Is this known, at least for ordinary Vandermonde matrices?


It seems
that positivity of $x\_1,x\_2,\dots $ cannot be dropped: chosing $e\_j$ integral
and the first few values of $x\_i$ negative leads in general to complex eigenvalues. 


Other remarks: (1) These matrices seem to be numerically highly instable, I have
to compute with very high accuracy (or work with Sturm sequences
over the integers). This limits the size of feasible cases somewhat.


(2) If all $e\_j$ are positive integers, the characteristic polynomial has alternating 
coefficients by the definition of Schur-polynomials as can be seen by
adapting David Speyer's answer to question 60938. Perhaps a clever continuity
argument (at least in the case $e\_1>0$) shows that the coefficients are always alternating?


If this observation is true, then there is perhaps a (more or less)
naturally associated symmetric matrix lurking behind.



Eigenvalues of generalized Vandermonde matrices
Expanding on my comment, the phenomenon that you are observing is explained by noting that:



> 
> The kernel $x^y$ is *strictly totally positive* (STP), i.e., for any choices of $0 < x\_1 < \cdots < x\_n$ and $y\_1 < y\_2 < \cdots < y\_n$ the matrix $K\_{ij} = x\_i^{y\_j}$ is (strictly) totally positive, i.e., all minors are positive. Since it is STP, its eigenvalues are also positive---for a proof see [Spectral Properties of Totally Positive Kernels and Matrices, in Total Positivity and its Applications, M. Gasca, C. A. Micchelli (eds.), pp. 477-511, Kluwer Academic Publishers, 1996](http://www2.math.technion.ac.il/~pinkus/papers/tp.pdf)
> 
> 
> 


For a proof of the STP property of this kernel, notice that it is essentially a special case of the $e^{xy}$ kernel (since $x\_i > 0$, we can write it as an exponent). The STP property of $e^{xy}$ is well-studied, and can be found for example in Pinkus's recent book (Chapter 4): [Totally Positive Matrices](http://rads.stackoverflow.com/amzn/click/0521194083).


