Convergence properties of additive and multiplicative Schwarz iterations for solving linear systems of equations with a symmetric positive semidefinite matrix are analyzed. The analysis presented applies to matrices whose principal submatrices are nonsingular, i.e., positive definite. These matrices appear in discretizations of some elliptic partial differential equations, e.g., those with Neum...

Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d imes d$ such that ${ m Tr}(A_kA_l)= $ for all $k,l=1,..., n$? For such matrices to exist, one must have $ geq 0$ for all $k,l=1,..., n$. We prove that if $n FRENKEL, Peter Erno, WEINER, Mihály. On vector configurations that can ...

Interior point methods can be extended to a number of cones (self-dual homogeneous cones) • Rn (linear programming) • vectorized symmetric matrices over real numbers (semidefinite programming) • vectorized Hermitian matrices over complex numbers • vectorized Hermitian matrices over quaternions • vectorized Hermitian 3×3 matrices over octonions Grötschel, Lovász and Schrijver [3]: semidefinite p...

In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a ...

A square matrix is said to be totally nonnegative (respectively, positive) if all of its minors are nonnegative (respectively, positive). Determinantal inequalities have been a popular and important subject, especially for positivity classes of matrices such as: positive semidefinite matrices, M−matrices, and totally nonnegative matrices. Our main interest lies in characterizing all of the ineq...

Notions of numerical ranges and joint numerical ranges of octonion matrices are introduced. Various properties of hermitian octonion matrices related to eigenvalues and convex cones, such as the convex cone of positive semidefinite matrices, are described. As an application, convexity of joint numerical ranges of 2×2 hermitian matrices is characterized. Another application involves existence of...

3 Operations and Properties 7 3.1 The Identity Matrix and Diagonal Matrices . . . . . . . . . . . . . . . . . . 8 3.2 The Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.5 Norms . . . ...

Notions of numerical ranges and joint numerical ranges of octonion matrices are introduced. Various properties of hermitian octonion matrices related to eigenvalues and convex cones, such as the convex cone of positive semidefinite matrices, are described. As an application, convexity of joint numerical ranges of 2×2 hermitian matrices is characterized. Another application involves existence of...

Curto and Fialkow proved in 1996 that flat positive semidefinite moment matrices always come from a finitely atomic positive measure. The tedious part of the proof is to show that flat moment matrices have always a flat extension. We give a new short argument for this based on Gröbner bases. Résumé. Curto et Fialkow ont démontré en 1996 que les matrices des moments, plates et semidéfinies posit...

Let A ∈ Mn(C ). We give a rank characterization of the semidefiniteness of Hermitian A in two ways. We show that A is semidefinite if and only if rank[X∗AX] = rank[AX], for all X ∈ Mn(C ), and we show that A is semidefinite if and only if rank[X∗AX] = rank[AXX∗], for all X ∈ Mn(C ). We show that if A has semidefinite Hermitian part and A has positive semidefinite Hermitian part then A satisfies...