Off-diagonal Entries Research Articles

Recovering a magnitude-symmetric matrix from its principal minors

We consider the inverse problem of finding a magnitude-symmetric matrix (matrix with opposing off-diagonal entries equal in magnitude) with a prescribed set of principal minors. This problem is closely related to the theory of recognizing and learning signed determinantal point processes in machine learning, as kernels of these point processes are magnitude-symmetric matrices. In this work, we prove a number of properties regarding sparse and generic magnitude-symmetric matrices. We show that principal minors of order at most ℓ, for some invariant ℓ depending only on principal minors of order at most two, uniquely determine principal minors of all orders. In addition, we produce a polynomial-time algorithm that, given access to principal minors, recovers a matrix with those principal minors using only a quadratic number of queries. Furthermore, when principal minors are known only approximately, we present an algorithm that approximately recovers a matrix, and show that the approximation guarantee of this algorithm cannot be improved in general.

Entries of the bottleneck matrices of maximal outerplanar graphs

In this paper, we consider the entries of the bottleneck matrices of maximal outerplanar graphs. We recall patterns from [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] involving the entries of bottleneck matrices for trees. We first establish similar patterns involving the diagonal entries of bottleneck matrices for maximal outerplanar graphs. We then investigate patterns involving the off-diagonal entries of bottleneck matrices for maximal outerplanar graphs and establish similar results to those found in [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] for trees. We close with examples illustrating our results.

Spectral arbitrariness for trees fails spectacularly

Given a graph G, consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted. In particular, we construct trees with multiplicity lists that require a unique spectrum, up to shifting and scaling. This represents the most extreme possible failure of spectral arbitrariness for a multiplicity list, and greatly extends all previously known instances of this phenomenon, in which only single linear constraints on the eigenvalues were observed.

Estimates of Green matrix entries of selfadjoint unbounded block Jacobi matrices

In a wide class of block Jacobi matrices, the norms of Green matrix (resolvent) entries are estimated. This estimate depends on the rate of growth of the norms of the off-diagonal entries and on the distance from the spectral parameter to the essential spectrum if the latter is nonempty. The sharpness of this estimate is shown by an example.

Optimal quantum speed for mixed states

The question of how fast a quantum state can evolve is considered. Using the definition of squared speed based on the Euclidean distance given in (Brody and Longstaff 2019 Phys. Rev. Res. 2 033127), we present a systematic framework to obtain the optimal speed of a d-dimensional system evolved unitarily under a time-independent Hamiltonian. Among the set of mixed quantum states having the same purity, the optimal state is obtained in terms of its purity parameter. We show that for an arbitrary d, the optimal state is represented by a X-state with an additional property of being symmetric with respect to the secondary diagonal. For sufficiently low purities for which the purity exceeds the purity of maximally mixed state I/d by at most 2/d2 , the only nonzero off-diagonal entry of the optimal state is ϱ1d , corresponding to the transition amplitude between two energy eigenstates with minimum and maximum eigenvalues, respectively. For larger purities, however, whether or not the other secondary diameter entries ϱi,d−i+1 take nonzero values depends on their relative energy gaps |Ed−i+1−Ei| . The effects of coherence and entanglement, with respect to the energy basis, are also examined and found that for optimal states both resources are monotonic functions of purity, so they can cause speed up quantum evolution leading to a smaller quantum speed limit. Our results show that although the coherence of the states is responsible for the speed of evolution, only the coherence caused by some off-diagonal entries located on the secondary diagonal play a role in the fastest states.

On the balanced truncation error bound and sign parameters from arrowhead realizations

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study.

Refined inertias of positive and hollow positive patterns

Abstract We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia ( n + , n − , n z , 2 n p ) \left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n + ≥ 1 {n}_{+}\ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n + ≥ 1 {n}_{+}\ge 1 and n − ≥ 2 {n}_{-}\ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order n n , we show that as n → ∞ n\to \infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for n n odd and 3/4 for n n even.

Time-constrained Gaussian mixture model for clustering multi-modal chemical process data

Two novel unsupervised learning algorithms were developed for improved clustering of multi-modal time series data that are not separable in feature space, which are common characteristics of chemical process data. The algorithms are extensions of the conventional Gaussian Mixture Model (GMM) and K-means clustering. Both algorithms were adapted to account for the time-dependent nature of chemical process data and thus are termed time-constrained GMM (TCGMM) and time-constrained K-means (TCK-means). The algorithms are evaluated using autoregressive time series data with small step changes in the means and variances; a problem that confounds conventional clustering algorithms. In Case Study 1, step changes in the means and variances are implemented at specific time intervals to create two modes. TCGMM outperforms the other algorithms by obtaining a minimum of 85% accuracy in correctly identifying the modes. The TCGMM algorithm is also tested in a second case study where combinations of mean- and variance-shifts are randomly instantiated based on a conditional probability table (CPT). TCGMM outperforms conventional GMM by an average accuracy of 65.4% versus 46.6% and learns the CPT with an average difference in the main diagonal entries (probabilities of remaining in the same mode) of 1.89% and an average difference in the off-diagonal entries (mode transition probabilities) of 0.664%. Keywords: clustering, TCGMM, TCK-means

Ideals of submaximal minors of sparse symmetric matrices

We study the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all (n−1)-minors of a symmetric n×n matrix whose entries in the upper triangle are distinct variables or zeros, and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded in a simple graph G with n vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of n and a single invariant of G. Moreover, it turns out that these ideals are always radical and have Cohen–Macaulay quotients if and only if G is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to G.

Morse theory for discrete magnetic operators and nodal count distribution for graphs

Given a discrete Schrödinger operator h on a finite connected graph G of n vertices, the nodal count \phi(h,k) denotes the number of edges on which the k -th eigenvector changes sign. A signing h' of h is any real symmetric matrix constructed by changing the sign of some off-diagonal entries of h , and its nodal count is defined according to the signing. The set of signings of h lie in a naturally defined torus \mathbb{T}_{h} of “magnetic perturbations” of h . G. Berkolaiko [Anal. PDE 6 (2013), 1213–1233] discovered that every signing h' of h is a critical point of every eigenvalue \lambda_{k}\colon \mathbb{T}_{h} \to \mathbb{R} , with Morse index equal to the nodal surplus. We add further Morse theoretic information to this result. We show if h_{\alpha}\in \mathbb{T}_{h} is a critical point of \lambda_{k} and the eigenvector vanishes at a single vertex v of degree d , then the critical point lies in a non-degenerate critical submanifold of dimension d+n-4 , closely related to the configuration space of a planar linkage. We compute its Morse index in terms of spectral data. The average nodal surplus distribution is the distribution of values of \phi(h',k)-(k-1) , averaged over all signings h' of h . If all critical points correspond to simple eigenvalues with nowhere-vanishing eigenvectors, then the average nodal surplus distribution is binomial. In general, we conjecture that the nodal surplus distribution converges to a Gaussian in a CLT fashion as the first Betti number of G goes to infinity.

Optimal Transport and Seismic Rays

We present a theoretical framework that links Fermat’s principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it yields shortest paths in the considered media through the optimal transport plans. In the discrete setting, our formulation results in physically significant optimal couplings, whose off-diagonal entries identify shortest paths in both directed and undirected graphs. For undirected graphs with positive edge weights, commonly used to parameterize seismic media, our method provides solutions to the Eikonal equation consistent with those from the Dijkstra algorithm. For directed negative-weight graphs, corresponding to transportation cost matrices with negative entries, our approach aligns with the Bellman–Ford algorithm but offers considerable computational advantages. We also highlight potential research directions. These include the use of sparse cost matrices to reduce the number of unknowns and constraints in the considered transportation problem, and solving specific classes of optimal transport problems through the Dijkstra algorithm to enhance computational efficiency.

Sparsification of large ultrametric matrices: insights into the microbial Tree of Life

Ultrametric matrices appear in many domains of mathematics and science; nevertheless, they can be large and dense, making them difficult to store and manipulate, unlike large but sparse matrices. In this manuscript, we exploit that ultrametric matrices can be represented as binary trees to sparsify them via an orthonormal base change based on Haar-like wavelets. We show that, with overwhelmingly high probability, only an asymptotically negligible fraction of the off-diagonal entries in random but large ultrametric matrices remain non-zero after the base change; and develop an algorithm to sparsify such matrices directly from their tree representation. We also identify the subclass of matrices diagonalized by the Haar-like wavelets and supply a sufficient condition to approximate the spectrum of ultrametric matrices outside this subclass. Our methods give computational access to a covariance matrix model of the microbiologists’ Tree of Life, which was previously inaccessible due to its size, and motivate introducing a new wavelet-based (beta-diversity) metric to compare microbial environments. Unlike the established metrics, the new metric may be used to identify internal nodes (i.e. splits) in the Tree that link microbial composition and environmental factors in a statistically significant manner.

Canonical systems whose Weyl coefficients have dominating real part

For a two-dimensional canonical system y′ (t) = zJH(t)y(t) on the half-line (0, ∞) whose Hamiltonian H is a.e. positive semi-definite, denote by qH its Weyl coefficient. De Branges’ inverse spectral theorem states that the assignment H ↦ qH is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions.The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re qH(iy), dominates its Poisson integral Im qH(iy) for y → +∞. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive M of H. It merely depends on the relative size of the off-diagonal entries of M compared with the diagonal entries. The second condition is of geometric nature and technically more complicated. It involves the relative size of the off-diagonal entries of H, a measurement for oscillations of the diagonal of H, and a condition on the speed and smoothness of the rotation of H.

Covariance structure estimation with Laplace approximation

The Gaussian covariance graph model is popular for revealing the underlying dependency structures among random variables. In this paper, we consider a spike and slab prior, which is a mixture of point-mass and normal distribution, on the off-diagonal entries. The spike and slab prior naturally introduces sparsity to the covariance structure so that the resulting posterior renders covariance structure learning. Under the spike and slab prior, we calculate the posterior model probabilities of covariance structures and natural Bayesian quantities for model selection using the Laplace approximation. We show that the error due to the Laplace approximation becomes asymptotically marginal at a rate that depends on the posterior convergence rate of the covariance matrix under the Frobenius norm. We propose a covariance structure estimation method based on the approximated posterior model probabilities. We also propose a block coordinate descent algorithm to determine the mode of the posterior density conditional on the structure of the covariance. The posterior mode is an estimate of the covariance matrix once the structure is chosen and the Laplace approximation is computed around it. Through a simulation study based on five numerical models, we demonstrate that the proposed method outperforms its competitors. The proposed method is applied to the breast cancer and Parkinson’s disease datasets, as well as the prediction of telephone call counts using telephone call center data, and compared with its competitors in terms of the linear discriminant analysis classification accuracy.

Positivity preservers forbidden to operate on diagonal blocks

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at “opposite ends”, and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.

Replica approach to the generalized Rosenzweig-Porter model

The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size NN of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.

Discrimination of geometrically uniform states and the no-signaling principle

The link between quantum state discrimination and the no-signaling principle is applied to discriminating geometrically uniform states. Specifically, the original discrimination problem is converted to constructing the complementary matrix with predetermined off-diagonal entries. Two such constructions are given explicitly, and therefore three upper bounds on the success probability are derived. Those upper bounds are then employed to estimate the robustness of coherence, where we establish a larger class of states with equal robustness of coherence and l 1-norm of coherence, and provide the exact answer of the robustness of coherence for any states that are diagonalized by Hadamard matrices.

Can the Flap of a Butterfly’s Wings Shift a Tornado into Texas—Without Chaos?

In our title, “chaos” means there is a positive Lyapunov exponent that causes the tornado to move. We are asserting that a positive Lyapunov exponent is not always needed to have a butterfly effect. Lorenz’s butterfly effect initially appeared in meteorology and has captured the imaginations of people for applications to all kinds of fields. We feel it is important to understand simpler non-meteorological models to understand the additional aspects of the butterfly effect. This paper presents simple linear map models that lack “chaos” but exhibit a butterfly effect: our simplest model does not have any positive Lyapunov exponents but still exhibits a butterfly effect, that is, temporary exponential growth from a tiny perturbation such as one infected mosquito setting off an epidemic outbreak. We focus on a 24-dimensional version of the map where a significant butterfly effect is observed even though the only Lyapunov exponent is 0. We introduce a linear “infected mosquito” model that shows how off-diagonal matrix entries can cause a finite-time growth rate. We argue that the degree of instability in our systems can be better measured by its finite-time growth rate. Our findings suggest that even in linear systems, off-diagonal matrix entries can significantly impact the system’s behavior and be more important than the Lyapunov exponents in higher-dimensional systems. A focus on finite-time growth rates can yield valuable insights into the system’s dynamics.

Centered PSD matrices with thin spectrum are M-matrices

We show that real, symmetric, centered (zero row sum) positive semidefinite matrices of order $n$ and rank $n-1$ with eigenvalue ratio $\lambda_{\max}/\lambda_{\min}\leq n/(n-2)$ between the largest and smallest nonzero eigenvalue have nonpositive off-diagonal entries, and that this eigenvalue criterion is tight. The result is relevant in the context of matrix theory and inverse eigenvalue problems, and we discuss an application to Laplacian matrices.

Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\Im q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for most Hamiltonians also $\Im q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations.