On intervals and sets of hypermatrices (tensors)

Abstract

Interval hypermatrices (tensors) are introduced and interval \alpha-hypermatrices are uniformly characterized using a finite set of 'extreme' hypermatrices, where \alpha can be strong P, semi-positive, or positive definite, among many others. It is shown that a symmetric interval is an interval (strictly) copositive-hypermatrix if and only if it is an interval (E) E₀-hypermatrix. It is also shown that an even-order, symmetric interval is an interval positive (semi-) definite-hypermatrix if and only if it is an interval P (P₀)-hypermatrix. Interval hypermatrices are generalized to sets of hyper-matrices, several slice-properties of a set of hypermatrices are introduced and sets of hypermatrices with various slice-properties are uniformly characterized. As a consequence, several slice-properties of a compact, convex set of hyper-matrices are characterized by its extreme points.

Keywords

Tensor / hypermatrix / interval hypermatrix / hypermatrix set / slice-P-property

References

[1] Bai X, Huang Z, Wang Y. Global uniqueness and solvability for tensor complementarity problems. J Optim Theory Appl, 2016, 170(2): 72–84

[2] Kostrikin A I, Manin Yu I. Linear Algebra and Geometry. Algebra, Logic and Applications Ser, Vol 1. Amsterdam: Gordon and Breach Science Publishers, 1997

[3] Landsberg J M. Tensors: Geometry and Applications. Grad Stud Math, Vol 128. Providence: Amer Math Soc, 2012

[4] Lim L H. Tensors and hypermatrices. In: Hogben L, ed. Handbook of Linear Algebra. 2nd ed. Boca Raton: CRC Press, 2013, 15-1–15-28

[5] Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40(6): 1302–1324

[6] Qi L. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439(1): 228–238

[7] Rockafellar R T. Convex Analysis. Princeton Math Ser, Vol 28. Princeton: Princeton Univ Press, 1970

[8] Rohn J, Rex G. Interval P-matrices. SIAM J Matrix Anal Appl, 1996, 17(4): 1020–1024

[9] Song Y, Gowda M S, Ravindran G. On some properties of P-matrix sets. Linear Algebra Appl, 1999, 290(1-3): 237–246

[10] Song Y S, Qi L. Necessary and sufficient conditions for copositive tensors. Linear Multilinear Algebra, 2015, 63(1): 120–131

[11] Song Y S, Qi L. Properties of some classes of structured tensors. J Optim Theory Appl, 2015, 165(3): 854–873

[12] Song Y S, Qi L. Tensor complementarity problem and semi-positive tensors. J Optim Theory Appl, 2016, 169(3): 1069–1078

[13] Tawhid M A, Rahmati S. Complementarity problems over a hypermatrix (tensor) set. Optim Lett, 2018, 12(6): 1443{1454

[14] Wilkinson J H. The Algebraic Eigenvalue Problem. Oxford: Clarendon Press, 1965

RIGHTS & PERMISSIONS

Higher Education Press