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### Current Research

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

Let $A$ be a square matrix of order $n$. If $A$ and $A^2$ are both P-matrices (matrices with all principal minors positive), then $A$ is positive stable, that is, the real part of every eigenvalue of $A$ 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 $A$ and the identity matrix $I$ is necessary and sufficient for all real eigenvalues of $A$ 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 $A$ and the corresponding rows of $I$ is equivalent to $A$ 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 $I$ is nonsingular?".

### Papers

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