Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley
TL;DR: In this article, the authors give a formal treatment of determinantal identities of the minors of a matrix and give a common, concise derivation of some important determinantial identities attributed to the mathematicians in the title.
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 141 citations till now. The article focuses on the topics: Determinantal point process.
Citations
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TL;DR: A concise survey on totally positive matrices and related topics can be found in this paper, where the authors present a unified method for computing the total positivity of a matrices.
443 citations
TL;DR: In this paper, the authors show that if a is regular, then the ratio of 1.1 (n
(1.1) √ n (1)
√ 1.
Abstract: $$\\left[ {\\begin{array}{*{20}{c}} {10} \\\\ { - v{{a}^{{ - 1}}}1} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {au} \\\\ {vb} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {1 - {{a}^{{ - 1}}}u} \\\\ {01} \\\\ \\end{array} } \\right] = \\left[ {\\begin{array}{*{20}{c}} {a0} \\\\ {0b - v{{a}^{{ - 1}}}u} \\\\ \\end{array} } \\right]$$
(1.1)
provided a is regular.
386 citations
TL;DR: A Toeplitz covariance matrix is presented that provides an approximation for power-law noise that is accurate for most GNSS time-series and shows a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time- series and the amount of missing data.
Abstract: One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires $$\mathcal{O }(n^3)$$
operations for $$n$$
observations. Previous research by the authors has shown that this amount of operations can be reduced to $$\mathcal{O }(n^2)$$
for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present a Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series.Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.
287 citations
Book•
05 May 2010TL;DR: In this article, the authors present a comprehensive and thorough account of the subject with careful treatment of the central properties of totally positive matrices, full proofs and a complete bibliography, with a tribute to the four people who have made the most notable contributions to the history of total positivity: I. J. Schoenberg, M. G. Krein, F. R. Gantmacher and S. Karlin.
Abstract: Totally positive matrices constitute a particular class of matrices, the study of which was initiated by analysts because of its many applications in diverse areas. This account of the subject is comprehensive and thorough, with careful treatment of the central properties of totally positive matrices, full proofs and a complete bibliography. The history of the subject is also described: in particular, the book ends with a tribute to the four people who have made the most notable contributions to the history of total positivity: I. J. Schoenberg, M. G. Krein, F. R. Gantmacher and S. Karlin. This monograph will appeal to those with an interest in matrix theory, to those who use or have used total positivity, and to anyone who wishes to learn about this rich and interesting subject.
197 citations
TL;DR: Schur complements as discussed by the authors are matrices of the form S = D − CaB, where a is a generalized inverse of A, and are the matrices for which the Schur complement of A in M = A B C D is the same as A in C D.
118 citations
References
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TL;DR: In this paper, the inverse of the sum of two matrices, one of them being nonsingular, has been studied and new expressions for the inverse were derived for the non-singular case.
Abstract: Available expressions are reviewed and new ones derived for the inverse of the sum of two matrices, one of them being nonsingular. Particular attention is given to $({\text{{\bf A}}} + {\text{{\bf ...
834 citations
Journal Article•
785 citations
TL;DR: In this paper, the authors show that if a is regular, then the ratio of 1.1 (n
(1.1) √ n (1)
√ 1.
Abstract: $$\\left[ {\\begin{array}{*{20}{c}} {10} \\\\ { - v{{a}^{{ - 1}}}1} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {au} \\\\ {vb} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {1 - {{a}^{{ - 1}}}u} \\\\ {01} \\\\ \\end{array} } \\right] = \\left[ {\\begin{array}{*{20}{c}} {a0} \\\\ {0b - v{{a}^{{ - 1}}}u} \\\\ \\end{array} } \\right]$$
(1.1)
provided a is regular.
386 citations