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Current Research

My current research stems from the following conjecture (made by Daniel Hershkowitz and Nathan Keller):

Let AA be a square matrix of order nn. If AA and A2A^2 are both P-matrices (matrices with all principal minors positive), then AA is positive stable, that is, the real part of every eigenvalue of AA is positive.

What fascinates me about this conjecture is the challenge that it presents despite the simplicity of its statement. It was originally conjectured (by Hershkowitz and Johnson in the mid-1980's) that matrices all of whose powers are P-matrices are positive stable. The stronger conjecture (by Hershkowitz and Keller above) was made after many unsuccessful attempts at constructing a counterexample.

Another part of my research includes generalizing the following fact:

The nonsingularity of all convex combinations of a square matrix AA and the identity matrix II is necessary and sufficient for all real eigenvalues of AA to be positive.

In the mid-1990's, Charles Johnson and Michael Tsatsomeros generalized this fact by proving that the nonsingularity of all convex combinations of the rows of AA and the corresponding rows of II is equivalent to AA being a P-matrix. I am currently working on generalizing this result within the class of interval matrices. The question that I am trying to answer is: "What additional spectral properties of A (aside from being a P-matrix) do we obtain if every convex combination of the entries of A and the corresponding entries of II is nonsingular?".


P. K. Torres, M. J. Tsatsomeros
Stability and Convex Hulls of Matrix Powers
Linear and Multilinear Algebra (to appear)