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Minors of the MoorePenrose Inverse ∗
, 1992
"... Let Qk,n = {α = (α1, · · · , αk) : 1 ≤ α1 < · · · < αk ≤ n} denote the strictly increasing sequences of k elements from 1,..., n. For α, β ∈ Qk,n we denote by A[α, β] the submatrix of A with rows indexed by α, columns by β. The submatrix obtained by deleting the αrows and βcolumns is de ..."
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× k minors of A † as a convex combination of the minors of A. The weights of this combination are, surprisingly, the same for all k. We apply our results to questions concerning the nonnegativity of principal minors of the MoorePenrose inverse.
or as a convex combination of MoorePenrose inverses of its maximal fullrank submatrices,
, 1990
"... Let A ∈ IRm×nr with nonzero singular values σ1, σ2, · · · , σr. The volume of A, volA, is defined as zero if r = 0, and otherwise, volA = r∏ i=1 σi, or equivalently, volA = det2AIJ, summing over all r×r nonsingular submatrices AIJ. The matrix volume volA generalizes the “absolute value of determi ..."
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of determinant ” from nonsingular to arbitrary matrices. Any rdimensional unit cube in R(AT) is mapped, under A, into a parallelepiped of volume volA. This paper gives properties and applications of volA. In particular, the MoorePenrose inverse of A is a convex combination of (ordinary) inverses of its maximal
Eigenvalues, invariant factors, highest weights, and Schubert calculus
 Bull. Amer. Math. Soc. (N.S
"... Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Gra ..."
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Cited by 176 (3 self)
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in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Contents 1. Eigenvalues of sums of Hermitian and real symmetric matrices 2. Invariant factors 3. Highest weights 4. Schubert calculus
Zeros of Gaussian Analytic Functions and Determinantal Point Processes
"... Key words and phrases. Gaussian analytic functions, zeros, determinantal processes, point processes, allocation, random matricesContents Preface vii ..."
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Cited by 53 (4 self)
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Key words and phrases. Gaussian analytic functions, zeros, determinantal processes, point processes, allocation, random matricesContents Preface vii
Solving Systems of Polynomial Equations
 AMERICAN MATHEMATICAL SOCIETY, CBMS REGIONAL CONFERENCES SERIES, NO 97
, 2002
"... One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, ..."
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Cited by 221 (14 self)
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One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture. The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the wellknown text books by Cox, Little, and O’Shea will find familiar themes in the first five chapters: polynomials in one variable, Gröbner
Gröbner bases and determinantal ideals
 COMMUTATIVE ALGEBRA, SINGULARITIES AND COMPUTER ALGEBRA
, 2003
"... We give an introduction to the theory of determinantal ideals and rings, their Gröbner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the KnuthRobinsonSchensted correspondence. The article contains a section treating the basic results about th ..."
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Cited by 5 (3 self)
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We give an introduction to the theory of determinantal ideals and rings, their Gröbner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the KnuthRobinsonSchensted correspondence. The article contains a section treating the basic results about
INFINITE DETERMINANTAL MEASURES AND THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES
"... ar ..."
Noncommutative desingularization of determinantal varieties
 I.” Inventiones Mathematicae
, 2010
"... In our paper “Noncommutative desingularization of determinantal varieties I”, we constructed and studied noncommutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction, we asserted that the results could be generalized to determinantal varieties de ..."
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Cited by 7 (1 self)
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In our paper “Noncommutative desingularization of determinantal varieties I”, we constructed and studied noncommutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction, we asserted that the results could be generalized to determinantal varieties
Results 1  10
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