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We show the existence of a broad class of affine Markov processes on the cone of positive self-adjoint Hilbert–Schmidt operators. Such processes are well-suited as infinite-dimensional stochastic covariance models. The class of processes we consider is an infinite-dimensional analogue of the affine processes on the cone of positive semi-definite and symmetric matrices studied in Cuchiero et al. (2011).As in the finite-dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well as jumps governed by a jump measure that depends affine linearly on the state. The fact that the cone of positive self-adjoint Hilbert–Schmidt operators has empty interior calls for a new approach to proving existence: instead of using standard localization techniques, we employ the theory on generalized Feller semigroups introduced in Dörsek and Teichmann (2010) and further developed in Cuchiero and Teichmann (2020). Our approach requires a second moment condition on the jump measures involved, consequently, we obtain explicit formulas for the first and second moments of the affine process.
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14On vector configurations that can be realized in the cone of positive matrices
On vector configurations that can be realized in the cone of positive matrices
1Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices
Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices
3Birkhoff-James orthogonality in the trace norm, with applications to quantum resource theories
Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.
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Measuring Sphericity in Positive Semi-Definite Matrices
The measure of sphericity for positive semi-definite matrices plays a crucial role in understanding their geometric properties, especially in high-dimensional settings. This paper introduces a robust measure of sphericity, which remains invariant under orthogonal transformations and scaling. We explore its behavior in finite-dimensional cases. Additionally, we investigate the stochastic case by considering a normal distribution, analyzing the asymptotic normality of random matrices and its implications on the convergence properties of the proposed measure.
Read moreThe Orthogonal Complement of Faces for Cones Associated with the Cone of Positive Semidefinite Matrices
It is known that the minimal cone for the constraint system of a conic linear optimization problem is a key component in obtaining strong duality without any constraint qualification. In the particular case of semidefinite optimization, an explicit expression for the dual cone of the minimal cone allows for a dual program of polynomial size that satisfies strong duality. This is achieved due to the fact that we can express the orthogonal complement of a face of the cone of positive semidefinite matrices completely in terms of a system of semidefinite inequalities. In this paper, we extend this result to cones that are either faces of the cone of positive semidefinite matrices or the dual cones of faces of the cone of positive semidefinite matrices. The newly proved result was used in Zhang (4OR 9:403–416, 2011). However, a proof was not given in Zhang (4OR 9:403–416, 2011).
Read more15Solving symmetric matrix word equations via symmetric space machinery
Solving symmetric matrix word equations via symmetric space machinery
15Sparse matrix decompositions and graph characterizations
Sparse matrix decompositions and graph characterizations
2Monotonicity of Positive Semidefinite Hermitian Matrices
Inequalities which compare elements of the convex cone of positive semidefinite hermitian matrices with products of roots of elements are proved. They yield inequalities for Schur functions (generalized matrix functions) which, when specialized to the determinant, give a result of R. Bellman and L. Mirsky.
Read more6Two orderings on a convex cone of nonnegative definite matrices
Two orderings on a convex cone of nonnegative definite matrices
128Positive semidefinite matrices with a given sparsity pattern
Positive semidefinite matrices with a given sparsity pattern
7Solvability of semidefinite complementarity problems
Solvability of semidefinite complementarity problems
1Improvements on some partial trace inequalities for positive semidefinite block matrices
We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove a celebrated result of Choi [Linear Algebra Appl. 516 (2017)]. The method also allows us to prove a generalization of another result of Choi [Linear Multilinear Algebra 66 (2018)]. Furthermore, we shall give an improvement on a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. In addition, we include with some majorization inequalities involving partial traces for two by two block matrices, and also provide inequalities related to the unitarily invariant norms as well as the singular values, which can be viewed as slight extensions of two results of Lin [Linear Algebra Appl. 459 (2014)] and [Electronic J Linear Algebra 31 (2016)].
Read more12Semidefinite programming lower bounds and branch-and-bound algorithms for the quadratic minimum spanning tree problem
Semidefinite programming lower bounds and branch-and-bound algorithms for the quadratic minimum spanning tree problem
7A New Algorithm for Positive Semidefinite Matrix Completion
Positive semidefinite matrix completion (PSDMC) aims to recover positive semidefinite and low-rank matrices from a subset of entries of a matrix. It is widely applicable in many fields, such as statistic analysis and system control. This task can be conducted by solving the nuclear norm regularized linear least squares model with positive semidefinite constraints. We apply the widely used alternating direction method of multipliers to solve the model and get a novel algorithm. The applicability and efficiency of the new algorithm are demonstrated in numerical experiments. Recovery results show that our algorithm is helpful.
Read more4The -positive semidefinite matrices A such that are nonsingular
This work focuses on the class of L -positive semidefinite matrices A such that A A ( L ) ( † ) − A ( L ) ( † ) A is nonsingular, where A ( L ) ( † ) is the generalized Bott-Duffin inverse of A with respect to a subspace L . We give some equivalent characterizations for the nonsingularity of A A ( L ) ( † ) − A ( L ) ( † ) A , A A ( L ) ( † ) + A ( L ) ( † ) A and I n − A ( A ( L ) ( † ) ) 2 A mainly by rank equalities and subspace operations, among which relations are pointed out. Meanwhile we show the inverses of their general versions in a decomposition form. In addition, the continuity of this class of matrices is discussed based on the continuity of the generalized Bott–Duffin inverse.
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