Refined Inertia of Matrix Patterns

Refined Inertia of Matrix Patterns

This paper explores how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. It demonstrates that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. A class of patterns is developed that are refined inertially arbitrary but not spectrally arbitrary, making use o...

Inertia of the Matrix

Let p1, . . . , pn be positive real numbers. It is well known that for every r < 0 the matrix [(pi + pj) r ] is positive definite. Our main theorem gives a count of the number of positive and negative eigenvalues of this matrix when r > 0. Connections with some other matrices that arise in Loewner’s theory of operator monotone functions and in the theory of spline interpolation are discussed.

A decomposition of the manipulator inertia matrix

Ela Refined Inertias of Tree Sign Patterns

The refined inertia (n+, n−, nz, 2np) of a real matrix is the ordered 4-tuple that subdivides the number n0 of eigenvalues with zero real part in the inertia (n+, n−, n0) into those that are exactly zero (nz) and those that are nonzero (2np). For n ≥ 2, the set of refined inertias Hn = {(0, n, 0, 0), (0, n − 2, 0, 2), (2, n − 2, 0, 0)} is important for the onset of Hopf bifurcation in dynamical...

Refined inertias of tree sign-patterns

The refined inertia (n+, n−, nz, 2np) of a real matrix is the ordered 4-tuple that subdivides the number n0 of eigenvalues with zero real part in the inertia (n+, n−, n0) into those that are exactly zero (nz) and those that are nonzero (2np). For n ≥ 2, the set of refined inertias Hn = {(0, n, 0, 0), (0, n − 2, 0, 2), (2, n − 2, 0, 0)} is important for the onset of Hopf bifurcation in dynamical...