Suppose Q(x) is a real n x n regular symmetric positive semidefinite matrix polynomial. Then it can be factored as Q(x) = G(x)TG(x), where G(x) is a real n x n matrix polynomial with degree half that of Q(x) if and on...
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Suppose Q(x) is a real n x n regular symmetric positive semidefinite matrix polynomial. Then it can be factored as Q(x) = G(x)TG(x), where G(x) is a real n x n matrix polynomial with degree half that of Q(x) if and only if det(Q(x)) is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equation XSX- XR +RTX + P = 0, where P, R, S are real n x n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization. (c) 2023 Elsevier Inc. All rights reserved.
Consider the following problem: given a positivesemidefinite (definite) matrix A is an element of Cmxm, and B is an element of C-nxm, such that parallel to((A)(B))parallel to(2 )= mu, find a matrix X satisfying paral...
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Consider the following problem: given a positivesemidefinite (definite) matrix A is an element of Cmxm, and B is an element of C-nxm, such that parallel to((A)(B))parallel to(2 )= mu, find a matrix X satisfying parallel to [GRAPHICS] parallel to(2 )= mu subject to [GRAPHICS] >= (>)0. In 1996, Zheng presented the expressions for X formed by Lowner partial ordering when the problem is solvable, and pointed out that there may be no X satisfying these expressions. In this paper, we present some sufficient and necessary conditions for the existence of X and provide its explicit general expressions.
Given an undirected graph G = (V, E) with node set V = [1, n], a subset S subset of or equal to V, and a rational vector a is an element ofQ(S boolean ORE), the positive semidefinite matrix completion problem consists...
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Given an undirected graph G = (V, E) with node set V = [1, n], a subset S subset of or equal to V, and a rational vector a is an element ofQ(S boolean ORE), the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n x n positive semidefinite matrix X = (x(ij)) satisfying x(ii) = a(i) (i is an element of S) and x(ij) = a(ij) (ij is an element of E). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in,the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time n the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length ( assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time n the real number model for the class of graphs containing no homeomorph of K-4.
This work explores the ratios of products of determinants of principal submatrices of positive definite matrices. We investigate conditions under which these ratios are bounded, particularly revisiting the necessary/s...
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This work explores the ratios of products of determinants of principal submatrices of positive definite matrices. We investigate conditions under which these ratios are bounded, particularly revisiting the necessary/sufficient conditions proposed by Johnson and Barrett. This analysis extends to set-theoretic consequences and unboundedness of certain ratios. We also demonstrate how these conditions can be used to prove the boundedness of several known determinantal inequalities. Additionally, we address the optimization problem of finding the supremum of such ratios over all positive definite matrices, formulating it as a linear optimization program. Finally, for completeness, we include the proofs of theorems that appear to have been previously known but lack accessible proofs. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
作者：
Al-Homidan, SulimanKFUPM
Dept Math Dhahran 31261 Saudi Arabia KFUPM
Ctr Smart Mobil & Logist Dhahran 31261 Saudi Arabia
The task of deducing directed acyclic graphs from observational data has gained significant attention recently due to its broad applicability. Consequently, connecting the log-det characterization domain with the set ...
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The task of deducing directed acyclic graphs from observational data has gained significant attention recently due to its broad applicability. Consequently, connecting the log-det characterization domain with the set of M-matrices defined over the cone of positive definite matrices has emerged as a crucial approach in this field. However, experimentally collected data often deviates from the expected positivesemidefinite structure due to introduced noise, posing a challenge in maintaining its physical structure. In this paper, we address this challenge by proposing four methods to reconstruct the initial matrix while maintaining its physical structure. Leveraging advanced techniques, including sequential quadratic programming (SQP), we minimize the impact of noise, ensuring the recovery of the reconstructed matrix. We provide a rigorous proof of convergence for the SQP method, highlighting its effectiveness in achieving reliable reconstructions. Through comparative numerical analyses, we demonstrate the effectiveness of our methods in preserving the original structure of the initial matrix, even in the presence of noise.
For i=1,...,k, let A(i) and B-i be positive definite matrices. It is shown thats (Sigma(m)(i=1)Ai#tBi)(R)0, p>0 and t is an element of[0,1]. This is stronger than the inequality |divided by((m)Sigma(i=1)A(i)(1/2)B(...
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For i=1,...,k, let A(i) and B-i be positive definite matrices. It is shown thats (Sigma(m)(i=1)Ai#tBi)(R)< log s((Sigma B-m(i=1)i)(tpr/2)(Sigma(m)(i=1)A(i))((1-t)pr)(Sigma B-m(i=1)i)(tpr/2))(1/p) for all r>0, p>0 and t is an element of[0,1]. This is stronger than the inequality |divided by((m)Sigma(i=1)A(i)(1/2)B(i)(1/2))(2)divided by divided by <=divided by divided by((m)Sigma(i=1)A(i))((m)Sigma B-i=1(i))divided by divided by, where A(i) commutes with B-i for each i and for all unitarily invariant norms, which has been proved by Audenaert. Applications of these inequalities shed some light on the solution of a question of Bourin.
Let A, B, and X be complex matrices of order n. We establish several novel singular value and norm inequalities, including: mu(k) (R(AX* B*)) <= root mu(k) (A vertical bar X vertical bar A* circle plus A vertical b...
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Let A, B, and X be complex matrices of order n. We establish several novel singular value and norm inequalities, including: mu(k) (R(AX* B*)) <= root mu(k) (A vertical bar X vertical bar A* circle plus A vertical bar X vertical bar A*) parallel to B vertical bar X*vertical bar B*parallel to <= = parallel to X parallel to parallel to B parallel to mu(k) (A circle plus A), mu(k) (AB * + BA*) <= mu(k) ((A* A + B* B) circle plus (AA* + BB*)), and for positivesemidefinite matrices A and B, mu(k) (AX - YB) <= max{parallel to A parallel to, parallel to B parallel to}(mu(k) (( X - Y) circle plus (X - Y))/2 + max{parallel to X parallel to, parallel to Y parallel to}). The first and second inequalities establish variants of well-known singular value inequalities, while the third result generalizes a commutator inequality for positivesemidefinite matrices previously established by Kittaneh. Additionally, we extend several classical singular value inequalities to functional settings, broadening their scope and applicability.
For n a positive integer let Q(1), Q(2) and Q(3) be n x n complex matrices. Then the 3 x 3 matrix M given by m(jk) = tr(vertical bar Q(j)(star)Q(k)vertical bar) is shown to be positivesemidefinite. (C) 2014 Elsevier ...
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For n a positive integer let Q(1), Q(2) and Q(3) be n x n complex matrices. Then the 3 x 3 matrix M given by m(jk) = tr(vertical bar Q(j)(star)Q(k)vertical bar) is shown to be positivesemidefinite. (C) 2014 Elsevier Inc. All rights reserved.
For a positive semidefinite matrix H = [A X X* B], we consider the norm inequality parallel to H parallel to <= parallel to A + B parallel to. We show that this inequality holds under certain conditions. Some relat...
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For a positive semidefinite matrix H = [A X X* B], we consider the norm inequality parallel to H parallel to <= parallel to A + B parallel to. We show that this inequality holds under certain conditions. Some related topics are also investigated. (C) 2019 Elsevier Inc. All rights reserved.
This paper is mainly concerned with solving the following two problems: Problem Ⅰ. Given X ∈ Rn×m, B . Rm×m. Find A ∈ Pn such thatwhereProblem Ⅱ. Given A ∈Rn×n. Find A ∈ SE such thatwhere F is Fro...
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This paper is mainly concerned with solving the following two problems: Problem Ⅰ. Given X ∈ Rn×m, B . Rm×m. Find A ∈ Pn such that
where
Problem Ⅱ. Given A ∈Rn×n. Find A ∈ SE such that
where F is Frobenius norm, and SE denotes the solution set of Problem I.
The general solution of Problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived.
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