Please use this identifier to cite or link to this item: http://hdl.handle.net/10603/444396
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dc.date.accessioned2023-01-12T12:14:00Z-
dc.date.available2023-01-12T12:14:00Z-
dc.identifier.urihttp://hdl.handle.net/10603/444396-
dc.description.abstractThe r-th Hadamard (or entrywise) power of a nonnegative positive semidefinite matrix need not be positive semidefinite for all positive real numbers r. The problems of studying the positive definiteness (or positive semidefiniteness) and total positivity (or total nonnegativity) of Hadamard powers of a matrix or a family of matrices have been of tremendous interest in matrix theory. An entrywise nonnegative matrix is called infinitely divisible if its r-th Hadamard power is positive semidefinite for every rgt0. For positive real numbers $\la_1lt\cdotslt\la_n$, we consider the matrix $\left[\frac{1}{\beta(\la_i,\la_j)}\right]$, where $\beta(.,.)$ denotes the beta function. We shall present our work on its infinite divisibility and the total positivity of its Hadamard powers. We shall give an important decomposition known as a successive elementary bidiagonal decomposition for the beta matrix $\mathcal{B}=\left[\frac{1}{\beta(i, j)}\right]$. We consider a few band matrices and identify all the Hadamard powers preserving the positive (semi) definiteness. We shall discuss similar results for a few more special matrices, namely, Bell matrices, Stirling matrices, characteristic matrices, and mean matrices. The matrix $S = [1+x_i y_j]$, where $0ltx_1lt\cdotsltx_n,\, 0lty_1lt\cdotslty_n$, has gained importance lately due to its role in powers preserving total nonnegativity. We shall give an explicit decomposition of $S$ in terms of elementary bidiagonal matrices. newline
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dc.languageEnglish
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dc.rightsuniversity
dc.titlePositivity properties of some special matrices and their Hadamard powers
dc.title.alternative
dc.creator.researcherSingh, Veer
dc.subject.keywordMathematics
dc.subject.keywordMathematics Applied
dc.subject.keywordPhysical Sciences
dc.description.note
dc.contributor.guideGrover, Priyanka and Reddy, A. Satyanarayana
dc.publisher.placeGreater Noida
dc.publisher.universityShiv Nadar University
dc.publisher.institutionDepartment of Mathematics
dc.date.registered2016
dc.date.completed2022
dc.date.awarded2022
dc.format.dimensions
dc.format.accompanyingmaterialDVD
dc.source.universityUniversity
dc.type.degreePh.D.
Appears in Departments:Department of Mathematics

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01_title.pdfAttached File59.8 kBAdobe PDFView/Open
02_prelim pages.pdf263.39 kBAdobe PDFView/Open
03_content.pdf86.51 kBAdobe PDFView/Open
04_introduction.pdf169.81 kBAdobe PDFView/Open
05_chapter 1.pdf197.5 kBAdobe PDFView/Open
06_chapter 2.pdf187.19 kBAdobe PDFView/Open
07_chapter 3.pdf115.57 kBAdobe PDFView/Open
08_chapter 4.pdf81.94 kBAdobe PDFView/Open
09_annexures.pdf56.86 kBAdobe PDFView/Open
80_recommendation.pdf90.22 kBAdobe PDFView/Open


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