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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?".

P. K. Torres, M. J. Tsatsomeros

Stability and Convex Hulls of Matrix PowersLinear and Multilinear Algebra (to appear)