If A is an element of R-mxn and B is an element of R-nxn, we define the product A circle divide B as A circle divide B = A circle times J(n) + J(m) circle times B, where circle times denotes the Kronecker product and ...
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If A is an element of R-mxn and B is an element of R-nxn, we define the product A circle divide B as A circle divide B = A circle times J(n) + J(m) circle times B, where circle times denotes the Kronecker product and J(n) is the n x n matrix of all ones. We refer to this product as the Cartesian product of A and B since if D-1 and D-2 are the distance matrices of graphs G(1) and G(2) respectively, then D-1 circle divide D-2 is the distance matrix of the Cartesian product G(1)square G(2). We study Cartesian products of Euclidean distance matrices (EDMs). We prove that if A and B are EDMs, then so is the product A circle divide B. We show that if A is an EDM and U is symmetric, then A circle times U is an EDM if and only if U = cJ(n) for some c. It is shown that for EDMs A and B, A circle divide B is spherical if and only if both A and B are spherical. If A and B are EDMs, then we derive expressions for the rank and the Moore Penrose inverse of A circle divide B. In the final section we consider the product A circle divide B for arbitrary matrices. For A is an element of R-mxm, we show that all nonzero minors of A circle divide B of order m + n - 1 are equal. An explicit formula for a nonzero minor of order m + n - 1 is proved. The result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n. (C) 2018 Elsevier Inc. All rights reserved.
We introduce and investigate the resolvent order, which is a binary relation on the set of firmly nonexpansive mappings. It unifies well-known orders introduced by Loewner (for positivesemidefinite matrices) and by Z...
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We introduce and investigate the resolvent order, which is a binary relation on the set of firmly nonexpansive mappings. It unifies well-known orders introduced by Loewner (for positivesemidefinite matrices) and by Zarantonello (for projectors onto convex cones). A connection with Moreaus order of convex functions is also presented. We also construct partial orders on (quotient sets of) proximal mappings and convex functions. Various examples illustrate our results.
Methods for solving the educational testing problem which arises from statistics are considered. The problem is to find lower bounds for the reliability of the total score on a test (or subtests) whose items are not p...
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Methods for solving the educational testing problem which arises from statistics are considered. The problem is to find lower bounds for the reliability of the total score on a test (or subtests) whose items are not parallel using data from a single test administration. We formulate the problem as an optimization problem with a linear objective function and semidefinite constraints. We maintain exact primal and dual feasibility during the course of the algorithm. The search direction is found using an inexact Gauss-Newton method rather than a Newton method on a symmetrized system. Computational results illustrating the robustness of the algorithm are successfully exploited.
A well-known characterization by Kraaijevanger [14] for Lyapunov diagonal stability states that a real, square matrix A is Lyapunov diagonally stable if and only if A o S is a P-matrix for any positivesemidefinite S ...
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A well-known characterization by Kraaijevanger [14] for Lyapunov diagonal stability states that a real, square matrix A is Lyapunov diagonally stable if and only if A o S is a P-matrix for any positivesemidefinite S with nonzero diagonal entries. This result is extended here to a new characterization involving similar Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. Among the main ingredients for this extension are a new concept called P-sets and a recent result regarding simultaneous Lyapunov diagonal stability by Berman, Goldberg, and Shorten [2]. (c) 2017 Elsevier Inc. All rights reserved.
We obtain expressions for the Moore-Penrose inverse of a Euclidean distance matrix (EDM) that are determined only by the positive semidefinite matrix associated with the EDM. The results complement formulas for the Mo...
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We obtain expressions for the Moore-Penrose inverse of a Euclidean distance matrix (EDM) that are determined only by the positive semidefinite matrix associated with the EDM. The results complement formulas for the Moore-Penrose inverse of an EDM given in Balaji and Bapat (2007) [2]. A formula for the inverse of a principal submatrix of an EDM is also derived, whose expression uses the Schur complement of the Laplacian of the EDM. As an application, we obtain an expression for the terminal Wiener index of a tree. (C) 2015 Elsevier Inc. All rights reserved.
Let A, B, and X be n x n complex matrices such that A and B are positivesemidefinite. If p,q > 1 with 1/p + 1/q = 1, it is shown that parallel to 1/p A(p) X + 1/q X B-q parallel to (2)(2) greater than or equal to ...
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Let A, B, and X be n x n complex matrices such that A and B are positivesemidefinite. If p,q > 1 with 1/p + 1/q = 1, it is shown that parallel to 1/p A(p) X + 1/q X B-q parallel to (2)(2) greater than or equal to 1/r2 parallel to A(p) X - X B-q parallel to(2)(2) + parallel to AXB parallel to(2)(2),where r = max(p, q) and parallel *** to(2) is the Hilbert-Schmidt norm. Generalizations and applications of this inequality are also considered. (C) 2000 Elsevier Science Inc. All rights reserved.
A matrix inequality of Thompson is extended. As an application, a partial affirmative answer to a recent question of Lin is provided. (C) 2017 Elsevier Inc. All rights reserved.
A matrix inequality of Thompson is extended. As an application, a partial affirmative answer to a recent question of Lin is provided. (C) 2017 Elsevier Inc. All rights reserved.
We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the properties that one would expect from a geometric mean, and...
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We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the properties that one would expect from a geometric mean, and that our geometric mean generalizes many inequalities satisfied by the geometric mean of two positivesemidefinite matrices. We prove some new properties of the geometric mean of two matrices, and give some simple computational formulae related to them for 2 x 2 matrices. (C) 2003 Elsevier Inc. All rights reserved.
We provide a new proof of a sharp lower bound for the rank of the Hadamard product of positivesemidefinite matrices. Our proof uses standard matrix analysis tools and involves the rank of one factor and the Kruskal r...
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We provide a new proof of a sharp lower bound for the rank of the Hadamard product of positivesemidefinite matrices. Our proof uses standard matrix analysis tools and involves the rank of one factor and the Kruskal rank of the other. It leads to a new understanding of when the Hadamard exponential of a positive semidefinite matrix can be singular. (C) 2020 Elsevier Inc. All rights reserved.
This letter proposes a constraint for a volume scattering power employing the principal minors, which can be used for polarimetric synthetic aperture radar (POLSAR) model-based decomposition. This constraint effective...
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This letter proposes a constraint for a volume scattering power employing the principal minors, which can be used for polarimetric synthetic aperture radar (POLSAR) model-based decomposition. This constraint effectively allows for avoiding unreasonable results which yield negative eigenvalues. The proposed constraint is derived so that all the principal minors of the coherency matrix after volume scattering subtraction are nonnegative. Mathematically, the constraint is exactly the same as that based on the nonnegative eigenvalues. Thus, it is guaranteed that the result is physically reasonable and that the volume scattering power is not overestimated. A significant advantage of the proposed method compared to the constraint based on the nonnegative eigenvalues is the high computation efficiency, since the maximal volume scattering power can be derived analytically, while the nonnegative eigenvalue constraint requires a numerical calculation. In our experiment, the computation of the maximal power is six times faster using the approach based on the principal minors than that based on the nonnegative eigenvalues.
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