Hadamard Functions of Inverse M-Matrices

Abstract

We prove that the class of GUM matrices is the largest class of bi-potential matrices stable under Hadamard increasing functions. We also show that any power α ≥ 1, in the sense of Hadamard functions, of an inverse M -matrix is also inverse Mmatrix showing a conjecture stated in Neumann [15]. We study the class of filtered matrices, which include naturally the GUM matrices, and present some sufficient conditions for a filtered matrix to be a bi-potential.

Inverse M-Matrices and Ultrametric Matrices

“…This was conjectured for r D 2 by Neumann in [54], and solved for integer r 1 by Chen in [11] and for general real numbers greater than 1 in [12]. We present the proof of this fact based on our paper [23]. As a complement to this result, we mention the characterization of matrices whose p-th Hadamard power is an inverse M -matrix, for large p, given in Johnson and Smith [39].…”

“…Hence, W .r/ is the inverse of an irreducible tridiagonal M -matrix. When W is a potential and r 1, the matrix W .r/ is also a potential (respectively Markov potential) see Theorem 2.2 (respectively Theorem 2.3) in [23].…”

“…We also study the conjecture performed by Neumann in [54] which states that the square function, in the sense of Hadamard, preserves the class of inverse M -matrices. This was solved for all integer powers by Chen in [11] and for general real powers greater than 1 in [12] (see also our article [23]). Our presentation relies on different ideas, where the concept of potential equilibrium plays a fundamental role.…”

Inverse M-matrix, a new characterization

“…In particular for the inverse M -problem we refer to the pioneer work of [8], [11] and [28]. Some results in the topic can be seen in [1], [2], [5], [6], [9], [10], [12], [15], [16], [17], and [27]. The special relation of this problem to ultrametric matrices in [4], [19], [20], [21], [23], [24], [25], [26].…”

Characterization of positively correlated squared Gaussian processes

“…Proof of Corollary 1.3. One can easily notice that (1) is equivalent to (3). Indeed, according to Bapat [1], a centered Gaussian vector (η i ) 1≤i≤n with nonsingular covariance matrix G is such that (η 2 i ) 1≤i≤n is infinitely divisible iff there exists a signature matrix σ [a diagonal matrix such that σ(i, i) = −1 or 1] such that σG −1 σ is a M -matrix (i.e., its offdiagonal entries are nonpositive).…”