Abstract
Let A be an m × n matrix with real entries. Given two proper cones K1 and K2 in ℝn and ℝm, respectively, we say that A is nonnegative if A(K1) ⊆ K2. A is said to be semipositive if there exists a \(x \in K_1^ \circ \) such that \(Ax \in K_2^ \circ \). We prove that A is nonnegative if and only if A + B is semipositive for every semipositive matrix B. Applications of the above result are also brought out.
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References
Chandrashekaran Arumugasamy, Sachindranath Jayaraman, and Vatsalkumar N. Mer, Semipositivity of linear maps relative to proper cones in finite dimensional real Hilbert spaces, Electronic Journal of Linear Algebra, 34 (2018), 304–319.
R. B. Bapat and T. E. S. Raghavan, Nonnegative matrices and applications, Encyclopedia of Mathematics and its Applications, 64, Cambridge University Press 1997.
A. Berman and R. J. Plemmons, Nonnegative matrices in the mathematical sciences, SIAM Classics in Applied Mathematics, Philadelphia, 1994.
B. Cain, D. Hershkowitz, and H. Schneider, Theorems of the alternative for cones and Lyapunov regularity of matrices, Czechoslovak Mathematical Journal, 47(122) (1997), 487–499.
J. Dieudonnè, Sur unè gènèralisation du groupe orthogonal à quatre variables [On a generalization that an orthogonal group has four variables], Archiv der Mathematik, 1(4) (1948/49), 282–287.
J. Dorsey, T. Gannon, N. Jacobson, C. R. Johnson, and M. Turnansky, Linear preservers of semi-positive matrices, Linear and Multilinear Algebra, 64(9) (2016), 1853–1862.
M. Seetharama Gowda and J. Tao, Z-transformations on proper and symmetric cones: Z-transformations, Mathematical Progremming, Series B, 117(1–2) (2009), 195–221.
M. Kasigwa and M. Tsatsomeros, Eventual cone invariance, Electronic Journal of Linear Algebra, 32 (2017), 204–216.
R. Loewy and H. Schneider, Positive operators on the n-dimensional ice cream cone, Journal of Mathematical Analysis and Applications, 49 (1975), 375–392.
K. C. Sivakumar and M. J. Tsatsomeros, Semipositive matrices and their semipositive cones, Positivity, 22(1) (2018), 379–398.
R. J. Stern and H. Wolkowicz, Exponential nonnegativity on the ice cream cone, SIAM Journal of Matrix Analysis and Applications, 12(1) (1991), 160–165.
B. S. Tam, On the structure of the cone of positive operators, Linear Algebra and its Applications, 167(1) (1992), 65–85.
Acknowledgement
The first author acknowledges the SERB, Government of India, for partial support in the form of a grant (Grant No. ECR/2017/000078). The third author acknowledges the Council of Scientific and Industrial Research (CSIR), India, for support in the form of Junior and Senior Research Fellowships (Award No. 09/997(0033)/2015-EMR-I).
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Arumugasamy, C., Jayaraman, S. & Mer, V.N. A Characterization of Nonnegativity Relative to Proper Cones. Indian J Pure Appl Math 51, 935–944 (2020). https://doi.org/10.1007/s13226-020-0442-4
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DOI: https://doi.org/10.1007/s13226-020-0442-4