Hadamard inverses, square roots and products of almost semidefinite matrices

Abstract Let A = (aij) be an n × n symmetric matrix with all positive entries and just one positive eigenvalue. Bapat proved then that the Hadamard inverse of A, given by A° (−1) = ( 1 a ij ) is positive semidefinite. We show that if moreover A is invertible then A°(−1) is positive definite. We use this result to obtain a simple proof that with the same hypotheses on A, except that all the diagonal entries of A are zero, the Hadamard square root of A, given by A° 1 2 = (a ij 1 2 ) , has just one positive eigenvalue and is invertible. Finally, we show that if A is any positive semidefinite matrix and B is almost positive definite and invertible then A ○ B ⪰ (1/ e T B −1 e )A .

[1]  R. Bapat.  Multinomial probabilities, permanents and a conjecture of Karlin and Rinott , 1988 .

[2]  P. Tarazaga,et al.  CIRCUM-EUCLIDEAN DISTANCE MATRICES AND FACES , 1996 .

[3]  I. J. Schoenberg,et al.  Metric spaces and positive definite functions , 1938 .

[4]  Jacques A. Ferland,et al.  Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons , 1982, Math. Program..

[5]  Miroslav Fiedler,et al.  An observation on the Hadamard product of Hermitian matrices , 1995 .

[6]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[7]  W. Donoghue.  Monotone Matrix Functions and Analytic Continuation , 1974 .

[8]  P. Lancaster,et al.  The theory of matrices : with applications , 1985 .

[9]  A note on the determinants and eigenvalues of distance matrices , 1989 .

[10]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[11]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[12]  Leonard M. Blumenthal,et al.  Theory and applications of distance geometry , 1954 .

[13]  I. J. Schoenberg.  On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space , 1937 .

[14]  Frank Critchley,et al.  The partial order by inclusion of the principal classes of dissimilarity on a finite set, and some of their basic properties , 1994 .

[15]  T. Raghavan,et al.  Nonnegative Matrices and Applications , 1997 .

[16]  John W. Auer.  An elementary proof of the invertibility of Distance matrices , 1995 .

[17]  J. Gower.  Euclidean Distance Geometry , 1982 .

[18]  Charles R. Johnson,et al.  Topics in matrix analysis: The Hadamard product , 1991 .

[19]  J. Ferland.  Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions , 1981 .

[20]  C. Ballantine.  On the Hadamard product , 1968 .

[21]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .