The Experts below are selected from a list of 309 Experts worldwide ranked by ideXlab platform
Yongge Tian - One of the best experts on this subject based on the ideXlab platform.
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extremal ranks of Submatrices in an hermitian solution to the matrix equation axa b with applications
Journal of Applied Mathematics and Computing, 2010Co-Authors: Yonghui Liu, Yongge TianAbstract:Suppose that AXA*=B is a consistent matrix equation and partition its Hermitian solution X*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the Submatrices in an Hermitian solution X to AXA*=B. From these formulas we derive necessary and sufficient conditions for the Submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.
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some properties of Submatrices in a solution to the matrix equation axb c with applications
Journal of The Franklin Institute-engineering and Applied Mathematics, 2009Co-Authors: Yongge TianAbstract:Abstract Suppose that AXB = C is a consistent matrix equation and partition its solution X into a 2×2 block form. In this article we give some formulas for the maximal and minimal ranks of the Submatrices in a solution X to AXB = C . From these formulas, we derive necessary and sufficient conditions for the Submatrices to be zero and nonsingular, respectively. As applications, we give a group of formulas for the maximal and minimal ranks of Submatrices in generalized inverses of matrices and their properties.
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Extremal ranks of Submatrices in an Hermitian solution to the matrix equation AXA*=B with applications
Journal of Applied Mathematics and Computing, 2009Co-Authors: Yonghui Liu, Yongge TianAbstract:Suppose that AXA*=B is a consistent matrix equation and partition its Hermitian solution X*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the Submatrices in an Hermitian solution X to AXA*=B. From these formulas we derive necessary and sufficient conditions for the Submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.
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The inverse of any two-by-two nonsingular partitioned matrix and three matrix inverse completion problems
Computers & Mathematics with Applications, 2009Co-Authors: Yongge Tian, Yoshio TakaneAbstract:A formula for the inverse of any nonsingular matrix partitioned into two-by-two blocks is derived through a decomposition of the matrix itself and generalized inverses of the Submatrices in the matrix. The formula is then applied to three matrix inverse completion problems to obtain their complete solutions.
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Some properties of Submatrices in a solution to the matrix equation AXB=C with applications
Journal of the Franklin Institute, 2009Co-Authors: Yongge TianAbstract:Abstract Suppose that AXB = C is a consistent matrix equation and partition its solution X into a 2×2 block form. In this article we give some formulas for the maximal and minimal ranks of the Submatrices in a solution X to AXB = C . From these formulas, we derive necessary and sufficient conditions for the Submatrices to be zero and nonsingular, respectively. As applications, we give a group of formulas for the maximal and minimal ranks of Submatrices in generalized inverses of matrices and their properties.
Yonghui Liu - One of the best experts on this subject based on the ideXlab platform.
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extremal ranks of Submatrices in an hermitian solution to the matrix equation axa b with applications
Journal of Applied Mathematics and Computing, 2010Co-Authors: Yonghui Liu, Yongge TianAbstract:Suppose that AXA*=B is a consistent matrix equation and partition its Hermitian solution X*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the Submatrices in an Hermitian solution X to AXA*=B. From these formulas we derive necessary and sufficient conditions for the Submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.
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Extremal ranks of Submatrices in an Hermitian solution to the matrix equation AXA*=B with applications
Journal of Applied Mathematics and Computing, 2009Co-Authors: Yonghui Liu, Yongge TianAbstract:Suppose that AXA*=B is a consistent matrix equation and partition its Hermitian solution X*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the Submatrices in an Hermitian solution X to AXA*=B. From these formulas we derive necessary and sufficient conditions for the Submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.
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Some properties of Submatrices in a solution to the matrix equations AX=C, XB=D
Journal of Applied Mathematics and Computing, 2008Co-Authors: Yonghui LiuAbstract:Suppose that AX=C, XB=D has a common solution and partition its solution \(X=\bigl[{\fontsize{7.5}{9}\selectfont \begin{array}{cc}X_{1}&X_{2}\\X_{3}&X_{4}\end{array}}\bigr]\) . In this paper, we give some formulas for the maximal and minimal ranks of the Submatrices in a solution X to matrix equations AX=C, XB=D. In addition, we investigate the uniqueness and the independence of Submatrices in a solutions X to this equations.
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Rank equalities for Submatrices in generalized inverse MT,S(2) of M
Applied Mathematics and Computation, 2004Co-Authors: Yonghui Liu, Musheng WeiAbstract:By applying the group inverse expressions of the generalized inverse M"T","S^(^2^) of the matrix M, we present some rank equalities for Submatrices in M"T","S^(^2^). As applications, we derive rank equalities for Submatrices in the Moore-Penrose inverse and the Drazin inverse.
Mohan Ravichandran - One of the best experts on this subject based on the ideXlab platform.
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Principal Submatrices, Restricted Invertibility, and a Quantitative Gauss–Lucas Theorem
International Mathematics Research Notices, 2018Co-Authors: Mohan RavichandranAbstract:Abstract We apply the techniques developed by Marcus, Spielman, and Srivastava, working with principal Submatrices in place of rank-$1$ decompositions to give an alternate proof of their results on restricted invertibility. This approach recovers results of theirs’ concerning the existence of well-conditioned column Submatrices all the way up to the so-called modified stable rank. All constructions are algorithmic. The main novelty of this approach is that it leads to a new quantitative version of the classical Gauss–Lucas theorem on the critical points of complex polynomials. We show that for any degree $n$ polynomial $p$ and any $c \geq 1/2$, the area of the convex hull of the roots of $p^{(\lfloor cn \rfloor )}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.
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PRINCIPAL Submatrices AND RESTRICTED INVERTIBILITY
arXiv: Functional Analysis, 2016Co-Authors: Mohan RavichandranAbstract:We slightly modify the techniques developed by Marcus, Spielman and Srivastava, working with principal Submatrices in place of rank $1$ decompositions to give an alternate proof of their results on restricted invertibility. We show that one can find well conditioned column Submatrices all the way upto the so called modified stable rank. All constructions are algorithmic.
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Principal Submatrices, restricted invertibility and a quantitative Gauss-Lucas theorem
2016Co-Authors: Mohan RavichandranAbstract:We apply the techniques developed by Marcus, Spielman and Srivastava, working with principal Submatrices in place of rank $1$ decompositions to give an alternate proof of their results on restricted invertibility. We show that one can find well conditioned column Submatrices all the way upto the so called modified stable rank. All constructions are algorithmic. A byproduct of these results is an interesting quantitative version of the classical Gauss-Lucas theorem on the critical points of complex polynomials. We show that for any degree $n$ polynomial $p$ and any $c \geq \frac{1}{2}$, the area of the convex hull of the roots of $p^{(cn)}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.
Guang-jing Song - One of the best experts on this subject based on the ideXlab platform.
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Ranks of Submatrices in (skew-)Hermitian solutions to a quaternion matrix equation
Journal of Applied Mathematics and Computing, 2012Co-Authors: Jing Jiang, Guang-jing SongAbstract:Assume that X and Y are Hermitian solutions to quaternion matrix equation AXA ∗+BYB ∗=C, which are partitioned into 2×2 block forms. We in this paper give the maximal and minimal ranks of Submatrices in the Hermitian solutions X=X ∗, Y=Y ∗, and establish necessary and sufficient conditions for the Submatrices to be zero, unique as well as independent. The findings of this paper widely extend the known results in the literature.
Andrew B. Nobel - One of the best experts on this subject based on the ideXlab platform.
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On the maximal size of Large-Average and ANOVA-fit Submatrices in a Gaussian Random Matrix
Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability, 2013Co-Authors: Xing Sun, Andrew B. NobelAbstract:We investigate the maximal size of distinguished Submatrices of a Gaussian random matrix. Of interest are Submatrices whose entries have an average greater than or equal to a positive constant, and Submatrices whose entries are well fit by a two-way ANOVA model. We identify size thresholds and associated (asymptotic) probability bounds for both large-average and ANOVA-fit Submatrices. Probability bounds are obtained when the matrix and Submatrices of interest are square and, in rectangular cases, when the matrix and Submatrices of interest have fixed aspect ratios. Our principal result is an almost sure interval concentration result for the size of large average Submatrices in the square case.
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On the maximal size of Large-Average and ANOVA-fit Submatrices in a Gaussian Random Matrix
arXiv: Statistics Theory, 2010Co-Authors: Xing Sun, Andrew B. NobelAbstract:We investigate the maximal size of distinguished Submatrices of a Gaussian random matrix. Of interest are Submatrices whose entries have average greater than or equal to a positive constant, and Submatrices whose entries are well-fit by a two-way ANOVA model. We identify size thresholds and associated (asymptotic) probability bounds for both large-average and ANOVA-fit Submatrices. Results are obtained when the matrix and Submatrices of interest are square, and in rectangular cases when the matrix Submatrices of interest have fixed aspect ratios. In addition, we obtain a strong, interval concentration result for the size of large average Submatrices in the square case. A simulation study shows good agreement between the observed and predicted sizes of large average Submatrices in matrices of moderate size.
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On the size and recovery of Submatrices of ones in a random binary matrix
Journal of Machine Learning Research, 2008Co-Authors: Xing Sun, Andrew B. NobelAbstract:Binary matrices, and their associated Submatrices of 1s, play a central role in the study of random bipartite graphs and in core data mining problems such as frequent itemset mining (FIM). Motivated by these connections, this paper addresses several statistical questions regarding Submatrices of 1s in a random binary matrix with independent Bernoulli entries. We establish a three-point concentration result, and a related probability bound, for the size of the largest square submatrix of 1s in a square Bernoulli matrix, and extend these results to non-square matrices and Submatrices with fixed aspect ratios. We then consider the noise sensitivity of frequent itemset mining under a simple binary additive noise model, and show that, even at small noise levels, large blocks of 1s leave behind fragments of only logarithmic size. As a result, standard FIM algorithms, which search only for Submatrices of 1s, cannot directly recover such blocks when noise is present. On the positive side, we show that an error-tolerant frequent itemset criterion can recover a submatrix of 1s against a background of 0s plus noise, even when the size of the submatrix of 1s is very small.
