scholarly journals Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Hiroshi Kurata ◽  
Ravindra B. Bapat
Keyword(s):  
Euclidean Distance ◽  
Symmetric Matrix ◽  
Distance Matrix ◽  
Positive Semidefinite ◽  
Complete Characterization ◽  
Symmetric Matrices ◽  
Euclidean Distance Matrix ◽  
Positive Semidefinite Matrices ◽  
Special Cases

AbstractBy a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D

scholarly journals The enhanced principal rank characteristic sequence over a field of characteristic 2

2017 ◽  
Vol 32 ◽  
pp. 273-290 ◽  
Author(s):  
Xavier Martínez-Rivera
Keyword(s):  
Symmetric Matrix ◽  
Complete Characterization ◽  
Symmetric Matrices ◽  
Characteristic Sequence ◽  
The Matrix ◽  
Characteristic 2

The enhanced principal rank characteristic sequence (epr-sequence) of an $n \times n$ symmetric matrix over a field $\F$ was recently defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either $\tt A$, $\tt S$, or $\tt N$ based on whether all, some (but not all), or none of the order-$k$ principal minors of the matrix are nonzero. Here, a complete characterization of the epr-sequences that are attainable by symmetric matrices over the field $\Z_2$, the integers modulo $2$, is established. Contrary to the attainable epr-sequences over a field of characteristic $0$, this characterization reveals that the attainable epr-sequences over $\Z_2$ possess very special structures. For more general fields of characteristic $2$, some restrictions on attainable epr-sequences are obtained.

scholarly journals Ad hoc microphone array calibration: Euclidean distance matrix completion algorithm and theoretical guarantees

2015 ◽  
Vol 107 ◽  
pp. 123-140 ◽  
Author(s):  
Mohammad J. Taghizadeh ◽  
Reza Parhizkar ◽  
Philip N. Garner ◽  
Hervé Bourlard ◽  
Afsaneh Asaei
Keyword(s):  
Euclidean Distance ◽  
Ad Hoc ◽  
Microphone Array ◽  
Matrix Completion ◽  
Distance Matrix ◽  
Euclidean Distance Matrix ◽  
Array Calibration ◽  
Euclidean Distance Matrix Completion

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

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