Inertia of some special matrices

Abstract

There have been several articles in literature that study the spectral behaviour of special classes of matrices such as the matrices based on power function given by the matrices , the matrices for positive values of r and positive real numbers . Bhatia and Jain in 2015 and Dyn, Goodman and Micchelli in 1986 have studied the spectral behaviour of and , respectively, for all real values of r. The power function is operator monotone when and operator convex when . It is natural to study the inertia of all these matrices when the power function is replaced by any operator monotone or operator convex function. In the present work, inertia of the matrices and is discussed for non-negative operator monotone and operator convex function f, which further leads to many known and new results.

Keywords:

• Conditionally positive definite matrices
• positive definite matrices
• infinitely divisible matrices
• inertia
• operator monotone functions
• operator convex functions

AMS Subject Classifications:

• 15A18
• 15A48
• 15A57
• 42A82

Acknowledgements

The authors thank Dr. Tanvi Jain for giving her suggestions for this work.

Notes

No potential conflict of interest was reported by the authors.