Inertia of non-integer Hadamard powers of a non-negative matrix. Issue 2 (1st February 2020)

• Record Type: Journal Article
• Authors: Garg, Isha; Aujla, Jaspal Singh
• Is Part Of: Linear & multilinear algebra. Volume 68:Issue 2(2020)
• Page Start: 410
• Page End: 416
• Publication Date: 2020-02-01
• Subjects: Conditionally negative definite matrix -- Hadamard power of a matrix -- matrix with one positive eigenvalue; 47A30 -- 47B15 -- 15A60; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1508405
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: We prove that if A = [ a i j ] is a conditionally negative definite matrix with all entries positive and f : ( 0, ∞ ) → ( 0, ∞ ) is a non-constant operator monotone function, then [ f ( a i j ) ] is also conditionally negative definite. Moreover, [ f ( a i j ) ] is invertible if A is invertible. Next, we prove that if A = [ a i j ] is a symmetric matrix with all entries positive and only one positive eigenvalue, and f ( t ) = t r for all t ∈ ( 0, ∞ ), r ∈ ( 0, 1 ], then [ f ( a i j ) ] also has only one positive eigenvalue and it is invertible if A is invertible. Analogous results are also considered when the diagonal entries of A are all zero and off-diagonal entries are all positive, which extends a result of Reams.