Chebyshev and Grüss inequalities for real rectangular matrices

Abstract

Quadratic ordering of rectangular real matrices implies Chebyshev’s inequality: If A1,…,Ar are m×n real matrices and B1,…,Br are n×q real matrices such that, for all i,j with 1⩽i,j⩽r, elementwise (Ai-Aj)(Bi-Bj)⩾0m×q, then for any real pj⩾0,j=1,…,r,∑jpj=1, elementwise (∑jpjAj)(∑jpjBj)⩽∑jpjAjBj. Further, linear ordering of rectangular real matrices implies Grüss’s inequality: If, elementwise, Aj⩽Aj+1,j=1,…,r-1 and elementwise Bj⩽Bj+1,j=1,…,r-1 then elementwise ∑jpjAjBj-(∑jpjAj)(∑jpjBj)⩽14(Ar-A1)(Br-B1). The bounds are sharp. These inequalities lead to inequalities for the spectral radius of nonnegative matrices. Linear ordering and quadratic ordering are equivalent for real scalars but not for real matrices.