Balanced matrices

In this paper we introduce a particular class of matrices. We study the concept of a matrix to be balanced. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix statistics in this setting. The crux will be to understanding the determinants and the eigen-values of balanced matrices. It turns out that there does exist a direct communication among the leading entry, the trace, determinants and, hence, the eigen-values of these matrices of order $2\times 2$. These matrices have an interesting property that enables us to predict their quadratic forms, even without knowing their entries but given their spectrum.Comment: 14 page

Balanced Sparsest Generator Matrices for MDS Codes

We show that given $n$ and $k$, for $q$ sufficiently large, there always exists an $[n, k]_q$ MDS code that has a generator matrix $G$ satisfying the following two conditions: (C1) Sparsest: each row of $G$ has Hamming weight $n - k + 1$; (C2) Balanced: Hamming weights of the columns of $G$ differ from each other by at most one.Comment: 5 page

An asymptotic existence result on compressed sensing matrices

For any rational number $h$ and all sufficiently large $n$ we give a deterministic construction for an $n\times \lfloor hn\rfloor$ compressed sensing matrix with $(\ell_1,t)$-recoverability where $t=O(\sqrt{n})$. Our method uses pairwise balanced designs and complex Hadamard matrices in the construction of $\epsilon$-equiangular frames, which we introduce as a generalisation of equiangular tight frames. The method is general and produces good compressed sensing matrices from any appropriately chosen pairwise balanced design. The $(\ell_1,t)$-recoverability performance is specified as a simple function of the parameters of the design. To obtain our asymptotic existence result we prove new results on the existence of pairwise balanced designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201

Algorithms for deterministic balanced subspace identification

New algorithms for identification of a balanced state space representation are proposed. They are based on a procedure for the estimation of impulse response and sequential zero input responses directly from data. The proposed algorithms are more efficient than the existing alternatives that compute the whole Hankel matrix of Markov parameters. It is shown that the computations can be performed on Hankel matrices of the input–output data of various dimensions. By choosing wider matrices, we need persistency of excitation of smaller order. Moreover, this leads to computational savings and improved statistical accuracy when the data is noisy. Using a finite amount of input–output data, the existing algorithms compute finite time balanced representation and the identified models have a lower bound on the distance to an exact balanced representation. The proposed algorithm can approximate arbitrarily closely an exact balanced representation. Moreover, the finite time balancing parameter can be selected automatically by monitoring the decay of the impulse response. We show what is the optimal in terms of minimal identifiability condition partition of the data into “past” and “future”