We study a natural random walk over the upper triangular matrices, with entries in the field Z2, generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields Zq for q prime.

It is well known that the generating function f ∈ L([−π, π],R) of a class of Hermitian Toeplitz matrices An(f) describes very precisely the spectrum of each matrix of the class [U. Grenader and G. Szegö, Toeplitz Forms and Their Applications, 2nd ed., Chelsea, New York, 1984; E. E. Tyrtyshnikov, Linear Algebra Appl., 232 (1996), pp. 1–43]. In this paper we consider n×n block Toeplitz matrices w...

This paper presents an approach for solving a nonlinear stochastic differential equations (NSDEs) using a new basis functions (NBFs). These functions and their operational matrices are used for representing matrix form of the NBFs. With using this method in combination with the collocation method, the NSDEs are reduced a stochastic nonlinear system of equations and unknowns. Then, the error ana...

The paper presents a subspace type of identification method for multivariable linear parameter-varying systems in state space representation with affine parameter dependence. It is shown that a major problem with subspace methods for this kind of systems is the enormous dimensions of the data matrices involved. To overcome the curse of dimensionality, we suggest to use only the most dominant ro...

$R_+^{n\times n}$ denotes the set of $n\times n$ non-negative matrices. For $A\in R_+^{n\times let $\Omega(A)$ be all matrices that can formed by permuting elements within each row $A$. Formally: $$\Omega(A)=\{B\in n}: \forall i\;\exists\text{ a permutation }\phi_i\; \text{s.t.}\ b_{i,j}=a_{i,\phi_i(j)}\;\forall j\}.$$ $B\in\Omega(A)$ $\rho(B)$ denote spectral radius or largest non negative eig...

A permuted graph matrix is a matrix V ∈ C(m+n)×m such that every row of the m×m identity matrix Im appears at least once as a row of V . Permuted graph matrices can be used in some contexts in place of orthogonal matrices, for instance when giving a basis for a subspace U ⊆ Cm+n, or to normalize matrix pencils in a suitable sense. In these applications the permuted graph matrix can be chosen wi...

Abstract. Monotone triangles are certain triangular arrays of integers, which correspond to n × n alternating sign matrices when prescribing (1, 2, . . . , n) as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved mo...

An Alternating sign matrix is a square matrix of 0’s, 1’s, and −1’s in which the sum of the entries in each row or column is 1 and the signs of the nonzero entries in each row or column alternate. This paper attempts to define an analogue to alternating sign matrices which is infinite and periodic. After showing the analogue we define shares desirable cahracteristics with alternating sign matri...

Let R=(r1, ..., rm) and S=(s1, ..., sn) be nonnegative integral vectors with ; ri=; sj. Let A(R, S) denote the set of all m×n {0, 1}-matrices with row sum vector R and column sum vector S. Suppose A(R, S) ]”. The interchange graph G(R, S) of A(R, S) was defined by Brualdi in 1980. It is the graph with all matrices in A(R, S) as its vertices and two matrices are adjacent provided they differ by ...