Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem

Call an $n$-by-$n$ invertible matrix $S$ a \emph{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 $\mathcal{C}(S) := \{ x \in \mathbb{R}^n: S D_x S^{-1} \geq 0,~D_x := \text{diag}(x) \}$ and $\mathcal{P})(S) := \{x \in \mathcal{C}(S) : x_1 = 1 \}$, which we call the \emph{Perron spectracone} and \emph{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 (\cite[Theorem 2.4]{f1974}) for Hadamard orders, and a constructive version of \cite[Theorem 5.1]{bh1991} for Sule\u{\i}manova spectra.Comment: To appear in Linear Algebra and its Application

Positive Eigenvalues of Generalized Words in Two Hermitian Positive Definite Matrices

We define a word in two positive definite (complex Hermitian) matrices $A$ and $B$ as a finite product of real powers of $A$ and $B$. The question of which words have only positive eigenvalues is addressed. This question was raised some time ago in connection with a long-standing problem in theoretical physics, and it was previously approached by the authors for words in two real positive definite matrices with positive integral exponents. A large class of words that do guarantee positive eigenvalues is identified, and considerable evidence is given for the conjecture that no other words do.Comment: 13 Pages, Novel Approaches to Hard Discrete Optimization, Fields Institute Communication

Necessary And Sufficient Conditions For Existence of the LU Factorization of an Arbitrary Matrix

If $A$ is an n-by-n matrix over a field $F$ ($A\in M_{n}(F)$), then $A$ is said to ``have an LU factorization'' if there exists a lower triangular matrix $L\in M_{n}(F)$ and an upper triangular matrix $U\in M_{n}(F)$ such that $$A=LU.$$ We give necessary and sufficient conditions for LU factorability of a matrix. Also simple algorithm for computing an LU factorization is given. It is an extension of the Gaussian elimination algorithm to the case of not necessarily invertible matrices. We consider possibilities to factors a matrix that does not have an LU factorization as the product of an ``almost lower triangular'' matrix and an ``almost upper triangular'' matrix. There are many ways to formalize what almost means. We consider some of them and derive necessary and sufficient conditions. Also simple algorithms for computing of an ``almost LU factorization'' are given

Symmetric Word Equations in Two Positive Definite Letters

A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B), as a map from the set of positive definite matrices into itself. Given positive definite P, B, and a symmetric word, S(A,B), with positive powers of A, we defne a symmetric word equation as an equation of the form S(A,B) = P. Such an equation is solvable if there is always a positive definite solution A for any given B and P. We prove that all symmetric word equations are solvable. Applications of this fact, methods for solution, questions about unique solvability (injectivity), and generalizations are also discussed.Comment: 9 page

On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices

It is shown that the polynomial $$p(t) = \text{Tr}[(A+tB)^m]$$ has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equivalent to $p(t)$ having positive coefficients for any $m$ and any two $n$-by-$n$ positive definite matrices. We show that, generally, the question in the real case reduces to that of singular $A$ and $B$, and this is a key part of our proof.Comment: 7 pages, J. Statistical Physic

Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup containing all bounded ratios is given. This allows the complete determination of the semigroup of bounded ratios for 4-by-4 positive definite matrices, whose 46 generators include ratios not implied by Hadamard-Fischer and ratios not bounded by 1. For n > 4 it is shown that the containment of semigroups is strict, but a generalization of nullity patterns, of which one example is given, is conjectured to provide a finite determination of all bounded ratios.Comment: 7 page

Eigenvalues of Words in Two Positive Definite Letters

The question of whether all words in two real positive definite letters have only positive eigenvalues is addressed and settled (negatively). This question was raised some time ago in connection with a long-standing problem in theoretical physics. A large class of words that do guarantee positive eigenvalues is identified, and considerable evidence is given for the conjecture that no other words do. In the process, a fundamental question about solvability of symmetric word equations is encountered.Comment: 13 pages, SIAM Journal of Matrix Analysi

The Proportion of Trees that are Linear

We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on $n$ vertices, characterize the asymptotic growth rate of the number of nonisomorphic linear trees, and show that the distribution of $k$-linear trees on $n$ vertices follows a central limit theorem.Comment: 8 pages; v2 contains new central limit theorem, as suggested by the refere

The critical exponent for continuous conventional powers of doubly nonnegative matrices

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than $n-2$ and we conjecture that it is always $n-2$ (as it is with Hadamard powering). We prove this conjecture when $n<6$ and in certain other special cases. We establish a quadratic bound for the critical exponent in general.Comment: 9 page

Positive eigenvalues and two-letter generalized words

A generalized word in two letters $A$ and $B$ is an expression of the form $W=A^{\alpha_1}B^{\beta_1}A^{\alpha_2}B^{\beta_2}... A^{\alpha_N}B^{\beta_N}$ in which the exponents $\alpha_i$, $\beta_i$ are nonzero real numbers. When independent positive definite matrices are substituted for $A$ and $B$, we are interested in whether $W$ necessarily has positive eigenvalues. This is known to be the case when N=1 and has been studied in case all exponents are positive by two of the authors. When the exponent signs are mixed, however, the situation is quite different (even for 2-by-2 matrices), and this is the focus of the present work.Comment: 6 Pages, Electronic Journal of Linear Algebr