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A Novel Method for Determining the Rank of a Matrix Adm, Mohammad


An $n$-by-$m$ Cauchon diagram $C$ is an $n$-by-$m$ grid consisting of $n∙m$ squares colored black and white, where each black square has the property that every square to its left (in the same row) or every square above it (in the same column) is black. Let $A=(a_{ij})$ be an $n$-by-$m$ matrix and $C$ an $n$-by-$m$ Cauchon diagram. Then we say that $A$ is a Cauchon matrix associated with the Cauchon diagram $C$ if for all $(i,j) \in \{1,…,n\} \times \{1,…,m\}$, we have $a_{ij}=0$ if and only if the corresponding square $(i,j)$ in $C$ is black. In this talk, we present a novel method for the determination of the rank of a matrix A and for checking a set of its consecutive row (or column) vectors for linear independence provided that the resulting matrix $\tilde{A}$ of the application of the condensed form of the Cauchon algorithm, see e.g., [2], is a Cauchon matrix. This method is also linked to the elementary bidiagonal factorization of a matrix under certain conditions [1]. This is joint work with Khawla Al Muhtaseb and Ayed Abdel Ghani (Palestine Polytechnic University, Hebron, Palestine), Shaun M. Fallat (University of Regina, Regina, Canada), and Juergen Garloff (University of Applied Sciences / HTWG Konstanz, and University of Konstanz, Konstanz, Germany). References: [1] M. Adm, K. Al Muhtaseb, A. Abedel Ghani, S. Fallat, and J. Garloff, A novel method for determining the rank of a matrix with application to bidiagonal factorization, submitted. [2] M. Adm and J. Garloff, Improved tests and characterizations of totally nonnegative matrices, Electron. J. Linear Algebra, 27, 588-610, 2014.

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