Related papers: On the distance from a weakly normal matrix polyno…

This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in "User-friendly tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both upper…

Given a normal matrix $A$ and an arbitrary square matrix $B$ (not necessarily of the same size), what relationships between $A$ and $B$, if any, guarantee that $B$ is also a normal matrix? We provide an answer to this question in terms of…

We give a lower bound for the value at q=1 of a Kazhdan-Lustig polynomial in a Weyl group W in terms of "patterns''. This is expressed by a "pattern map" from W to W' for any parabloic subgroup W'. This notion generalizes the concept of…

We consider the bifurcation problem u'' + \lambda u = N(u) with two point boundary conditions where N(u) is a general nonlinear term which may also depend on the eigenvalue \lambda. A new derivation of a variational principle for the lowest…

Let $\Lambda$ be the limiting smallest eigenvalue in the general (\beta, a)-Laguerre ensemble of random matrix theory. Here \beta>0, a >-1; for \beta=1,2,4 and integer a, this object governs the singular values of certain rank n Gaussian…

The spectral $p$-norm of $r$-matrices generalizes the spectral $2$-norm of $2$-matrices. In 1911 Schur gave an upper bound on the spectral $2$-norm of $2$-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to $r$-matrices.…

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,\delta)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $\delta$. Previous works introduced this concept and study when…

We derive computable formulas for the structured backward errors of a complex number $\lambda$ when considered as an approximate eigenvalue of rational matrix polynomials that carry a symmetry structure. We consider symmetric,…

In a recent paper, Bruns and von Thaden established a bound for the length of vectors involved in a unimodular triangulation of simplicial cones. The bound is exponential in the square of the logarithm of the multiplicity, and improves…

The paper studies upper bounds for the total variation distance between two polynomials of a special form in random vectors satisfying the Doeblin-type condition. Our approach is based on the recent results concerning Nikolskii--Besov-type…

In this paper we bring to light an unprecedented property of the eigenvalues of a matrix A with the eigenvalues and eigenvectors of a submatrix of A. This property can be used, through the technique developed here, to determine some of…

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by $ 2 \*\sigma + o(N^{-6/11+\epsilon}), $ where $\sigma^2 $ is the…

Let $A$ be a $n\times n$ complex Hermitian matrix and let $\lambda(A)=(\lambda_1,\ldots,\lambda_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising…

We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in…

The current work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix $M_n = M + N_n$ where $M$ is deterministic, symmetric with large operator norm and $N_n$ is a random symmetric matrix with…

We prove quadratic eigenvalue perturbation bounds for generalized Hermitian eigenvalue problems. The bounds are proportional to the square of the norm of the perturbation matrices divided by the gap between the spectrums. Using the results…

We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B…

The Davis-Kahan theorem can be used to bound the distance of the spaces spanned by the first $r$ eigenvectors of any two symmetric matrices. We extend the Davis-Kahan theorem to apply to the comparison of the union of eigenspaces of any two…

The luminosity distance in the standard cosmology as given by $\Lambda$CDM and consequently the distance modulus for supernovae can be defined by the Pad\'e approximant. A comparison with a known analytical solution shows that the Pad\'e…

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…