共查询到20条相似文献,搜索用时 15 毫秒
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A determinantal identity, frequently used in the study of totally positive matrices, is extended, and then used to re-prove the well-known univariate knot insertion formula for B-splines. Also we introduce a class of matrices, intermediate between totally positive and strictly totally positive matrices. The determinantal identity is used to show any minor of such matrices is positive if and only if its diagonal entries are positive. Among others, this class of matrices includes B-splines collocation matrices and Hurwitz matrices.This author acknowledges a sabbatical stay at IBM T.J. Watson Research Center in 1990, which was supported by a DGICYT grant from Spain. 相似文献
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An n×m real matrix A is said to be totally positive (strictly totally positive) if every minor is nonnegative (positive). In this paper, we study characterizations of these classes of matrices by minors, by their full rank factorization and by their thin QR factorization. 相似文献
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For any given set S of n distinct positive numbers, we construct a symmetric n-by-n (strictly) totally positive matrix whose spectrum is S. Thus, in order to be the spectrum of an n-by-n totally positive matrix, it is necessary and sufficient that n numbers be positive and distinct. 相似文献
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The class of Bernstein-Vandermonde matrices (a generalization of Vandermonde matrices arising when the monomial basis is replaced by the Bernstein basis) is considered. A convenient ordering of their rows makes these matrices strictly totally positive. By using results related to total positivity and Neville elimination, an algorithm for computing the bidiagonal decomposition of a Bernstein-Vandermonde matrix is constructed. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate. One of the applications of our algorithm is the design of fast and accurate algorithms for solving Lagrange interpolation problems when using the Bernstein basis, an approach useful for the field of Computer Aided Geometric Design since it avoids the stability problems involved with basis transformations between the Bernstein and the monomial bases. A different application consists of the use of the bidiagonal decomposition as an intermediate step of the computation of the eigenvalues and the singular value decomposition of a totally positive Bernstein-Vandermonde matrix. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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We prove certain interlacing inequalities for the eigenvalues of totally positive matrices. 相似文献
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Konstanze Rietsch 《Proceedings of the American Mathematical Society》1997,125(9):2565-2570
Given a complex reductive linear algebraic group split over with a fixed pinning, it is shown that all elements of the Lie algebra infinitesimal to the totally positive subsemigroup of lie in the totally positive cone .
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Lj. Arambašić 《Linear and Multilinear Algebra》2019,67(8):1685-1700
In this paper we establish a connection between full spark frames and totally nonsingular matrices. Then we provide a method for constructing infinite totally positive matrices which make up a subclass of the class of totally nonsingular matrices. Using this method we then construct a family of infinite totally positive matrices parameterized by non-negative numbers which contains, as the simplest case, the infinite Pascal matrix. The paper ends with some examples and comments on full spark frames. 相似文献
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O.Y. Kushel 《Linear algebra and its applications》2011,435(11):2722-2730
A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. Classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented. 相似文献
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Miroslav Fiedler 《Czechoslovak Mathematical Journal》2016,66(3):597-602
A close relationship between the class of totally positive matrices and anti-Monge matrices is used for suggesting a new direction for investigating totally positive matrices. Some questions are posed and a partial answer in the case of Vandermonde-like matrices is given. 相似文献
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A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these
minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive
matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape
preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal
characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations.
Both authors were partially supported by the DGICYT Spain Research Grant PB93-0310 相似文献
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A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations. 相似文献
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Colin W. Cryer 《Linear algebra and its applications》1973,7(1):83-92
An n × n real matrix A is an STP (strictly totally positive) matrix if all its minors are strictly positive. An n × n real triangular matrix A is a ΔSTP matrix if all its nontrivial minors are strictly positive. It is proved that A is an STP matrix iff A = LU where L is a lower triangular matrix, U is an upper triangular matrix, and both L and U are ΔSTP matrices. Several related results are proved. These results lead to simple proofs of many of the determinantal properties of STP matrices. 相似文献
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A new “finite section” type theorem is used to show that the members of an interesting class of bounded totally positive matrices map l∞ onto l∞ if and only if their range contains a vector which alternates in sign and has coordinates bounded away from zero. The class of matrices studied contains all banded totally positive matrices, and thus all infinite spline collocation matrices. Connections to related work and extension to matrices which are not sign regular are indicated. 相似文献
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In this paper, we analyze and characterize the cone of nonsymmetric positive semidefinite matrices (NS-psd). Firstly, we study basic properties of the geometry of the NS-psd cone and show that it is a hyperbolic but not homogeneous cone. Secondly, we prove that the NS-psd cone is a maximal convex subcone of P0-matrix cone which is not convex. But the interior of the NS-psd cone is not a maximal convex subcone of P-matrix cone. As the byproducts, some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given. Finally, we present some properties of metric projection onto the NS-psd cone. 相似文献
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For a strictly totally positive M × N matrix A we show that the ratio has exactly R = min{ M, N} nonzero critical values for each fixed p? (1, ∞). Letting λi denote the ith critical value, and xi an associated critical vector, we show that λ1 > … > λR > 0 and xi (unique up to multiplication by a constant) has exactly i ? 1 sign changes. These critical values are generalizations to lp of the s-numbers of A and satisfy many of the same extremal properties enjoyed by the s-numbers, but with respect to the lp norm. 相似文献
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