The goal of this paper is to prove the Central Limit Theorem for linear statistics of the eigenvalues of real symmetric band random matrices with independent entries. First, we define a real symmetric band random matrix. Let {bn} be a sequence of integers satisfying 0 ≤ bn ≤ n/2 such that bn → ∞ as n → ∞. Define dn(j, k) := min{|k − j|, n− |k − j|}, (1.1) In := {(j, k) : dn(j, k) ≤ bn, j, k = 1...

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those with a Hilbert–Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetr...

We present some observations on the block triangular form (btf) of structurally symmetric, square, sparse matrices. If the matrix is structurally rank deficient, its canonical btf has at least one underdetermined and one overdetermined block. We prove that these blocks are transposes of each other. We further prove that the square block of the canonical btf, if present, has a special fine struc...

Let D denote a positive integer and let QD denote the graph of the D-dimensional hypercube. Let X denote the vertex set of QD and let A ∈ MatX(R) denote the adjacency matrix of QD. A matrix B ∈ MatX(R) is called A-like whenever both (i) BA = AB; (ii) for all x, y ∈ X that are not equal or adjacent, the (x, y)-entry of B is zero. Let L denote the subspace of MatX(R) consisting of the A-like elem...

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated sym...

(P ) s.t. Ax = b, 0 ≤ x ≤ u, where the symmetric positive semidefinite matrix Q ∈ Rn×n, the rank m matrix A ∈ Rm×n, c ∈ Rn, u ∈ Rn and b ∈ Rm are the data for the problem, and x ∈ Rn is the vector of variables. Three aspects of this topic are explored. The first concerns a property of the Cholesky factorization of the normal equation matrix that arises at each iteration of an IPM applied to pro...

Over any field F every square matrix A can be factored into the product of two symmetric matrices as A = S1 ·S2 with Si = S i ∈ F and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius’ symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such ...

It is well-known how any symmetric matrix can be transformed into a similar tridiagonal one [1, 2]. This orthogonal similarity transformation forms the basic step for various algorithms. For example if one wants to compute the eigenvalues of a symmetric matrix, one can rst transform it into a similar tridiagonal one and then compute the eigenvalues of this tridiagonal matrix. Very recently an a...

In this paper, we study the largest and the smallest singular vectors of the generalized Lyapunov operator. For real matrices A,B with order n , we prove that max‖X‖F =1 ‖AXBT + BXA‖F is achieved by a symmetric matrix for n 3 and give a counterexample for order n = 4 . We also prove that min‖X‖F =1 ‖AXBT +BXA‖F is achieved by a symmetric matrix for n 2 and give a counterexample for order n = 3 ...

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated sym...