Pietro Paparella

Matrix Roots of Eventually Positive Matrices

Abstract Eventually positive matrices are real matrices whose powers become and remain strictly positive. As such, eventually positive matrices are a fortiori matrix roots of positive matrices, which motivates us to study the matrix roots of primitive matrices. Using classical matrix function theory and Perron–Frobenius theory, we characterize, classify, and describe in terms of the real Jordan canonical form the p th-roots of eventually positive matrices.

Perron spectratopes and the real nonnegative inverse eigenvalue problem

Abstract Call an n-by-n invertible matrix S a Perron similarity if there is a real non-scalar diagonal matrix D such that S D S − 1 is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra C ( S ) : = { x ∈ R n : S D x S − 1 ≥ 0 , D x : = diag ( x ) } and P ( S ) : = { x ∈ C ( S ) : x 1 = 1 } , which we call the Perron spectracone and Perron spectratope, respectively. The set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest. The Perron spectracone and spectratope of Hadamard matrices are of particular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows. In addition, we provide a constructive version of a result due to Fiedler [9, Theorem 2.4] for Hadamard orders, and a constructive version of the Boyle–Handelman theorem [2, Theorem 5.1] for Suleĭmanova spectra.

Matrix roots of imprimitive irreducible nonnegative matrices

Abstract Using matrix function theory, Perron–Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix p th-roots of imprimitive irreducible nonnegative matrices. Preliminary results concerning the matrix roots of reducible matrices are provided as well.

A Short and Elementary Proof of the Two-Sidedness of the Matrix Inverse

Summary We give an elementary proof of the two-sidedness of the matrix inverse using only linear independence and the reduced row-echelon form of a matrix, concepts prominent in an elementary linear algebra course. In addition, we show that a matrix is invertible if and only if it is row-equivalent to the identity matrix without appealing to elementary matrices.

Jordan chains of h-cyclic matrices

Abstract Arising from the classification of the matrix-roots of a nonnegative imprimitive irreducible matrix, we present results concerning the Jordan chains of an h -cyclic matrix. We also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples.

On the realizability of the critical points of a realizable list

Abstract The nonnegative inverse eigenvalue problem (NIEP) is to characterize the spectra of entrywise nonnegative matrices. A finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix. Monov conjectured that the kth-moments of the list of critical points of a realizable list are nonnegative. Johnson further conjectured that the list of critical points must be realizable. In this work, Johnsons conjecture, and consequently Monovs conjecture, is established for a variety of important cases including Ciarlet spectra, Suleĭmanova spectra, spectra realizable via companion matrices, and spectra realizable via similarity by a complex Hadamard matrix. Additionally we prove a result on differentiators and trace vectors, and use it to provide an alternative proof of a result due to Malamud and a generalization of a result due to Kushel and Tyaglov on circulant matrices. Implications for further research are discussed.

A matricial view of the Karpelevič Theorem

Abstract The question of the exact region in the complex plane of the possible single eigenvalues of all n-by-n stochastic matrices was raised by Kolmogorov in 1937 and settled by Karpelevic in 1951 after a partial result by Dmitriev and Dynkin in 1946. The Karpelevic result is unwieldy, but a simplification was given by Đokovic in 1990 and Ito in 1997. The Karpelevic region is determined by a set of boundary arcs each connecting consecutive roots of unity of order less than n. It is shown here that each of these arcs is realized by a single, somewhat simple, parameterized stochastic matrix. Other observations are made about the nature of the arcs and several further questions are raised. The doubly stochastic analog of the Karpelevic region remains open, but a conjecture about it is amplified.

The critical exponent for generalized doubly nonnegative matrices

It is known that the critical exponent (CE) for conventional, continuous powers of n-by-n doubly nonnegative (DN) matrices is . Here, we consider the larger class of diagonalizable, entrywise nonnegative n-by-n matrices with nonnegative eigenvalues (generalized doubly nonnegative (GDN)). We show that, again, a CE exists and is able to bind with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for . There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also being for DN matrices.

Row cones, perron similarities, and nonnegative spectra

Abstract In further pursuit of the diagonalizable real nonnegative inverse eigenvalue problem (RNIEP), we study the relationship between the row cone and the spectracone of a Perron similarity S. In the process, a new kind of matrix, row Hadamard conic (RHC), is defined and related to the D-RNIEP. Characterizations are given when , and explicit examples are given for all possible set-theoretic relationships between the two cones. The symmetric NIEP is the special case of the D-RNIEP in which the Perron similarity S is also orthogonal.

MATRIX FUNCTIONS THAT PRESERVE THE STRONG PERRON-FROBENIUS PROPERTY ∗

In this note, we characterize matrix functions that preserve the strong Perron-Frobenius property using the real Jordan canonical form of a real matrix.