qi = 1, pA ≥ v ≥ Aq. The number v is known as the value of game. The pair (p, q) is known as an equilibrium for the matrix game. As usual, x ≥ 0 means that every entry of the vector x is ≥ 0. We write y ≤ t for a vector y and a number t if every entry of y is ≤ t. We go even further in abusing notation, denoting by y− t the vector obtaining from y by subtracting t from every entry. Similarly we...

The paper investigates linear functionals ϕ : ℝn → ℝ preserving a set ℳ⊆ ℝ, i.e., such that ϕ(v) ∈ ℳ for any vector u with components in ℳ. For various types of subsets real numbers, characterizations the preserve them are obtained. In particular, sets ℤ,ℚ, ℤ+,ℚ+,ℝ+, several infinite integers, bounded and unbounded intervals, finite numbers considered. A characterization also allows one to desc...

In this paper, we give the factorizations of Lucas and inverse matrices. We also investigate Cholesky factorization symmetric matrix. Moreover, obtain upper lower bounds for eigenvalues matrix by using some majorization techniques.