Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

The Solution to a Structured Matrix Approximation Problem Using Grassman Coordinates

Abstract: A method for finding the best approximation of a matrix A by a full rank Hankel matrix is given. The initial problem of best approximation of one matrix by another is transformed to a problem involving best approximation of a given vector by a second vector whose elements are constrained so that its inverse image is a Hankel matrix. The map from a matrix to a vector is the invertible map between a subspace represented as the row space of the matrix A and the Grassman vector representing that subspace. The relation between the principle angles associated with a pair of subspaces and the angle between the Grassman vectors associated with the subspaces is established.

Modelling and estimation of mutual coupling between array elements

Abstract: The adverse effect of mutual coupling between the elements of an array, on the performance of super-resolution techniques is demonstrated. An analytical modelling of the mutual coupling is presented, and its effects are modelled in the form of a complex square matrix, the mutual coupling matrix (MCM). The MCM depends on the array geometry and the array electrical characteristics, but not on the direction of the incoming signals. Based on the modelling of the mutual coupling effects, a new technique is proposed for estimating the mutual coupling matrix of an array of general geometry. The proposed method employs one pilot source, and uses an extra element at some distance away from the original array. >

Identity-by-Descent Matrix Decomposition Using Latent Ancestral Allele Models

Abstract: Genetic linkage and association studies are empowered by proper modeling of relatedness among individuals. Such relatedness can be inferred from marker and/or pedigree information. In this study, the genetic relatedness among n inbred individuals at a particular locus is expressed as an n × n square matrix Q. The elements of Q are identity-by-descent probabilities, that is, probabilities that two individuals share an allele descended from a common ancestor. In this representation the definition of the ancestral alleles and their number remains implicit. For human inspection and further analysis, an explicit representation in terms of the ancestral allele origin and the number of alleles is desirable. To this purpose, we decompose the matrix Q by a latent class model with K classes (latent ancestral alleles). Let P be an n × K matrix with assignment probabilities of n individuals to K classes constrained such that every element is nonnegative and each row sums to 1. The problem then amounts to approximating Q by PPT, while disregarding the diagonal elements. This is not an eigenvalue problem because of the constraints on P. An efficient algorithm for calculating P is provided. We indicate the potential utility of the latent ancestral allele model. For representative locus-specific Q matrices constructed for a set of maize inbreds, the proposed model recovered the known ancestry.