On matrices with $Q^2$-scalings
We provide a counterexample to some statements dealing with a sufficient property for the square of a matrix to be a $P_0^+$ -matrix.
math.SP↗
arXiv subjects
O. Y. Kushel
Publications and source records attributed to O. Y. Kushel.
Generalized even and odd totally positive matrices
A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. The classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented.
math.SP↗
Cone-theoretic generalization of total positivity
This paper is devoted to the generalization of the theory of total positivity. We say that a linear operator A in R^n is generalized totally positive (GTP), if its jth exterior power preserves a proper cone K_j in the corresponding space for every j = 1, ..., n. We also define generalized strictly totally positive (GSTP) operators. We prove that the spectrum of a GSTP operator is positive and simple, moreover, its eigenvectors are localized in special sets. The existence of invariant cones of finite ranks is shown under some additional conditions. Some new insights and alternative proofs of the well-known results of Gantmacher and Krein describing the properties of TP and STP matrices are presented.
math.SP↗
Generalized Brouncker's continued fractions and their logarithmic derivatives
In this paper, we study the continued fraction y(s,r) which satisfies the equation y(s,r)y(s+2r,r)=(s+1)(s+2r-1) for r > 1/2. This continued fraction is a generalization of the Brouncker's continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s,r). The asymptotic series for y(s,r) at the infinity are also studied. The generalizations of some Ramanujan's formulas are presented.
math.NT↗
Spectral properties of one class of sign-symmertic matrices
A $n\times n$ matrix $A$, which has a certain sign-symmetric structure ($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix is similar to a nonnegative matrix. The existence of the second in modulus positive eigenvalue $λ_2$ of a $J$--sign-symmetric matrix $A$, or an odd number $k$ of simple eigenvalues, which coincide with the $k$-th roots of $ρ(A)^k$, is proved under the additional condition that its second compound matrix is also $J$--sign-symmetric. The conditions when a $J$--sign-symmetric matrix with a $J$--sign-symmetric second compound matrix has complex eigenvalues, which are equal in modulus to $ρ(A)$, are given.
math.SP↗