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Pietro Paparella

Publications and source records attributed to Pietro Paparella.

At least 19 recordsLinked to original sources
arXiv2026

On a conjecture concerning totally extremal ideal Perron similarities

Identifying ideal Perron similarities is a problem of central interest in the longstanding nonnegative inverse eigenvalue problem (NIEP). A normalized ideal Perron similarity is called totally extremal if every entry has modulus one. Recently, Gershnik et al. [J. Algebra 694 (2026), 782--800] proved that the character table of a finite Abelian group is totally extremal and conjectured the converse. In this paper, we settle this conjecture in the affirmative by first showing that the rows of a totally extremal normalized ideal Perron similarity form a group under the Hadamard product. Then, it is shown that the rows of a nonsingular matrix form a group under the Hadamard product if and only if it is the character table of a finite Abelian group, and we further show that this group is isomorphic to the underlying group. These results extend the classical theorem due to Romanovsky and Karpelevi\v{c} on the unimodular eigenvalues of stochastic matrices in the complex unit disk to the setting of spectratopes in the unit ball of complex Euclidean space.

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arXiv2025

Character tables are ideal Perron similarities

An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Each Perron similarity gives a nontrivial polyhedral cone, called the spectracone, and polytope, called the spectratope, of realizable spectra (thought of as vectors in complex Euclidean space). A Perron similarity is called ideal if its spectratope coincides with the conical hulls of its rows. Identifying ideal Perron similarities is of great interest in the pursuit of the longstanding nonnegative inverse eigenvalue problem. In this work, it is shown that the character table of a finite group is an ideal Perron similarity. In addition to expanding ideal Perron similarities to include a broad class of matrices, the results unify previous works into a single, theoretical framework. It is demonstrated that the spectracone can be described by finitely-many group-theoretic inequalities. When the character table is real, we derive a group-theoretic formula for the volume of the projected Perron spectratope, which is a simplex. Finally, an implication for further research is given.

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arXiv2024

Perron similarities and the nonnegative inverse eigenvalue problem

The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevich arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of $4 \times 4$ stochastic matrices. Similar to the Karpelevich region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. Extremal elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the discrete Fourier transform (DFT) matrix corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices.

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arXiv2024

Demystifying the Karpelevic theorem

The statement of the Karpelevic theorem concerning the location of the eigenvalues of stochastic matrices in the complex plane (known as the Karpelevic region) is long and complicated and his proof methods are, at best, nebulous. Fortunately, an elegant simplification of the statement was provided by Ito -- in particular, Ito's theorem asserts that the boundary of the Karpelevic region consists of arcs whose points satisfy a polynomial equation that depends on the endpoints of the arc. Unfortunately, Ito did not prove his version and only showed that it is equivalent. More recently, Johnson and Paparella showed that points satisfying Ito's equation belong to the Karpelevic region. Although not the intent of their work, this initiated the process of proving Ito's theorem and hence the Karpelevic theorem. The purpose of this work is to continue this effort by showing that an arc appears in the prescribed sector. To this end, it is shown that there is a continuous function $λ:[0,1] \longrightarrow \mathbb{C}$ such that $\mathsf{P}^\mathsf{I}(λ(α)) = 0$, $\forall α\in [0,1]$, where $\mathsf{P}^\mathsf{I}$ is a Type I reduced Ito polynomial. It is also shown that these arcs are simple. Finally, an elementary argument is given to show that points on the boundary of the Karpelevic region are extremal whenever $n > 3$.

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arXiv2024

Polynomials that preserve nonnegative monomial matrices

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we provide a formula for computing an arbitrary power of a monomial matrix and a formula for computing the polynomial of a nonnegative monomial matrix.

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arXiv2023

Matrices whose field of values is inscribed in a polygon

In this work, it is shown that if $A$ is an $n$-by-$n$ convexoid matrix (i.e., its field of values coincides with the convex hull of its eigenvalues), then the field of any $(n-1)$-by-$(n-1)$ principal submatrix of $A$ is inscribed in the field of $A$, i.e., the field is tangent to every side of the polygon corresponding to the boundary of the field of $A$. This result generalizes a special case established by Johnson and Paparella [Amer. Math. Monthly 127 (2020), no. 1,45-53].

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arXiv2023

The converse of the Cowling--Obrechkoff--Thron theorem

In this work, the converse of the Cowling--Obrechkoff--Thron theorem is established. In addition to its theoretical interest, the result fills a gap in the proof of Kellogg's celebrated eigenvalue inequality for matrices whose principal minors are positive or nonnegative.

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arXiv2022

Polynomials that preserve nonnegative matrices of order two

A known characterization for entire functions that preserve all nonnegative matrices of order two is shown to characterize polynomials that preserve nonnegative matrices of order two. Equivalent conditions are derived and used to prove that $\mathscr{P}_3 \subset \mathscr{P}_2$, which was previously unknown. A new characterization is given for polynomials that preserve nonnegative circulant matrices of order two.

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arXiv2021

Polynomials that preserve nonnegative matrices

In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83--90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this work, novel results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed.

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arXiv2021

Kronecker products of Perron similarities

An invertible matrix is called a Perron similarity if one of its columns and the corresponding row of its inverse are both nonnegative or both nonpositive. Such matrices are of relevance and import in the study of the nonnegative inverse eigenvalue problem. In this work, Kronecker products of Perron similarities are examined and used to to construct ideal Perron similarities all of whose rows are extremal.

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arXiv2021

Jordan chains of $h$-cyclic matrices, II

McDonald and Paparella [Linear Algebra Appl. 498 (2016), 145--159] gave a necessary condition on the structure of Jordan chains of $h$-cyclic matrices. In this work, that necessary condition is shown to be sufficient. As a consequence, we provide a spectral characterization of nonsingular, $h$-cyclic matrices. In addition, we provide results for the Jordan chains corresponding to the eigenvalue zero of singular matrices. Along the way, a new characterization of circulant matrices is given.

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arXiv2020

Matricial Proofs of Some Classical Results about Critical Point Location

The Gauss--Lucas and Bôcher--Grace--Marden theorems are classical results in the geometry of polynomials. Proofs of the these results are available in the literature, but the approaches are seemingly different. In this work, we show that these theorems can be proven in a unified theoretical framework utilizing matrix analysis (in particular, using the field of values and the differentiator of a matrix). In addition, we provide a useful variant of a well-known result due to Siebeck.

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arXiv2018

Realizing Suleĭmanova spectra via permutative matrices, II

In this work, the real nonnegative inverse eigenvalue problem is solved for a particular class of permutative matrix. The necessary and sufficient condition there is also shown to be sufficient for the symmetric nonnegative inverse eigenvalue problem. A result due to Johnson and Paparella [MR3452738, Linear Algebra Appl. 493 (2016), 281--300] is extended to include normalized lists that satisfy the new sufficient condition.

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arXiv2018

A proof of the elliptical range theorem via Kippenhahn's theorem

The elliptical range theorem asserts that the field of values (or numerical range) of a two-by-two matrix with complex entries is an elliptical disk, the foci of which are the eigenvalues of the given matrix. Many proofs of this result are available in the literature, but most, with one exception, are computational and quite involved. In this note, it is shown that the elliptical range theorem follows from the properties of plane algebraic curves and a straightforward application of a well-known result due to Kippenhahn.

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arXiv2018

Eisenstein's criterion, Fermat's last theorem, and a conjecture on powerful numbers

Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $μ$ is a zero of $X_n$ if and only if the triple $(μ,μ+m,μ+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the irreducibility of these polynomials implies Fermat's last theorem. It is also shown, in a precise asymptotic sense, that for a vast majority of cases, these polynomials are irreducible via Eisenstein's criterion. We conclude by offering a conjecture on powerful numbers.

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arXiv2017

On the realizability of the critical points of a realizable list

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 k\textsuperscript{th}-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, Johnson's conjecture, and consequently Monov's conjecture, is established for a variety of important cases including Ciarlet spectra, Sule\uımanova 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 alternate 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.

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