Matrix Roots of Eventually Positive Matrices

Eventually positive matrices are real matrices whose powers become and remain strictly positive. As such, eventually positive matrices are a fortiori matrix roots of positive matrices, which motivates us to study the matrix roots of primitive matrices. Using classical matrix function theory and Perron-Frobenius theory, we characterize, classify, and describe in terms of the real Jordan canonical form the $p$th-roots of eventually positive matrices.Comment: Accepted for publication in Linear Algebra and its Application

Extended M-matrices and subtangentiality

AbstractThe concept of a singular M-matrix A with respect to a proper cone K is extended, by replacing the usual regularity condition A = αI − B for a K-nonnegative matrix B with the weaker condition, exponential nonnegativity of - A. As in earlier work which dealt with the nonsingular case, in the present characterizations the lack of regularity is overcome by employing subtangentiality

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

summary:Let $a$, $b$ and $c$ be fixed complex numbers. Let $M_n(a,b,c)$ be the $n\times n$ Toeplitz matrix all of whose entries above the diagonal are $a$, all of whose entries below the diagonal are $b$, and all of whose entries on the diagonal are $c$. For $1\leq k\leq n$, each $k\times k$ principal minor of $M_n(a,b,c)$ has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of $M_n(a,b,c)$. We also show that all complex polynomials in $M_n(a,b,c)$ are Toeplitz matrices. In particular, the inverse of $M_n(a,b,c)$ is a Toeplitz matrix when it exists

Principal pivot transforms: properties and applications

AbstractThe principal pivot transform (PPT) is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right-hand side of the system. The notion of the PPT is encountered in mathematical programming, statistics and numerical analysis among other areas. The purpose of this paper is to draw attention to the main properties and uses of PPTs, make some new observations and motivate further applications of PPTs in matrix theory. Special consideration is given to PPTs of matrices whose principal minors are positive