Rang des matrices hermitiennes semi-definies positives indefiniment divisibles (Q1239216)

In this paper we study \(n \times n\) hermitian semi definite positive matrices \(A= (a(i,j))\) which are infinitely divisible in \textit{R. A. Horn's} sense [Trans. Am. Math. Soc. 136, 269--286 (1969; Zbl 0177.05003)], i.e. such that it is possible to find an \(n\times n\) hermitian matrix \(U = (u(i,j))\) satisfying: i) \(a (i,j) = e^{u (i,j)}\) if \(a (i,j) \neq 0\); ii) for all \(\theta > 0\), \(A_\theta = (a_\theta (i,j))\) is hermitian semi definite positive where \(A_\theta\) is defined by \(a_\theta (i,j) = e^{\theta u (i,j)}\) if \(a (i,j) \neq 0\); \(a_\theta (i, j) =0\) if \(a (i, j) =0\). The main result we prove is: ``All those matrices \(A_\theta\) (\(\theta >0\)), called the ``Hadamard power matrices of \(A\)'' have the same rank''. In addition, we give a representation theorem for matrices which are infinitely divisible in \(\mathbb R\) (such that the elements \(a_\theta (i, j)\) can be chosen real).