The minimal nonnegative solution G of the matrix equation G = A 0 + A 1 G + A 2 G 2 , where the matrices A 0 , A 1 and A 2 are nonnegative and A 0 + A 1 + A 2 is stochastic, plays an important role in the study of quasi-birth-death processes (QBDs). The cyclic reduction algorithm is a very efficient iteration for finding the matrix G, under the standard assumption that the transition probabilit...

In this paper, it is shown that nearly completely decomposable (NCD) Markov chains are quasi{lumpable. The state space partition is the natural one, and the technique may be used to compute lower and upper bounds on the stationary probability of each NCD block. In doing so, a lower bounding nonnegative coupling matrix is employed. The nature of the stationary probability bounds is closely relat...

 It is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. In this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.  

We show that an n-dimensional Riemannian manifold with n-nonnegative or n-nonpositive curvature operator of the second kind has restricted holonomy SO(n) is flat. The result does not depend on completeness and can be improved provided space Einstein Kähler. In particular, if a locally symmetric kind, then it constant curvature. When irreducible this to 3n2n+2n+4-nonnegative 3n2n+2n+4-nonpositiv...

In this paper, the nonnegative Q-matrix completion problem is studied. A real n × n matrix is a Q-matrix if for k ∈ {1, . . . , n}, the sum of all k × k principal minors is positive. A digraph D is said to have nonnegative Q-completion if every partial nonnegative Q-matrix specifying D can be completed to a nonnegative Q-matrix. For nonnegative Q-completion problem, necessary conditions and suf...

is known to be stable if and only if ρ(A) < 1. Models of real world dynamical phenomena often involve positive quantities. A dynamical system (1) is called positive if any trajectory of the system starting in the positive orthant R+ remains in R+. In this case, the matrix A has only real positive entries. In many cases, it may be useful to consider systems with a known “nominal” part A and a un...

We consider an adversarially-trained version of the nonnegative matrix factorization, a popular latent dimensionality reduction technique. In our formulation, attacker adds arbitrary bounded norm to given data matrix. design efficient algorithms inspired by adversarial training optimize for dictionary and coefficient matrices with enhanced generalization abilities. Extensive simulations on synt...

We consider the nonsymmetric algebraic Riccati equation XM12X + XM11 + M22X + M21 = 0, where M11,M12,M21,M22 are real matrices of sizes n × n, n × m,m × n, m × m, respectively, and M = [Mij ]i,j=1 is an irreducible singular M matrix with zero row sums. The equation plays an important role in the study of stochastic fluid models, where the matrix −M is the generator of a Markov chain. The soluti...