منابع مشابه
Loewner Matrices and Operator Convexity
Let f be a function from R+ into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form [ f(pi)−f(pj) pi−pj ] are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f(t) = tg(t) for some operator convex function g if and only if these matrices are conditio...
متن کاملPositivity and Conditional Positivity of Loewner Matrices
We give elementary proofs of the fact that the Loewner matrices [ f(pi)−f(pj) pi−pj ] corresponding to the function f(t) = t on (0,∞) are positive semidefinite, conditionally negative definite, and conditionally positive definite, for r in [0, 1], [1, 2], and [2, 3], respectively. We show that in contrast to the interval (0,∞) the Loewner matrices corresponding to an operator convex function on...
متن کاملThe Inertia of Hermitian Tridiagonal Block Matrices
Let H be a partitioned tridiagonal Hermitian matrix. We characterized the possible inertias of H by a system of linear inequalities involving the orders of the blocks, the inertia of the diagonal blocks and the ranks the lower and upper subdiagonal blocks. From the main result can be derived some propositions on inertia sets of some symmetric sign pattern matrices.
متن کاملModifying the inertia of matrices arising in optimization
Applications in constrained optimization (and other areas) produce symmetric matrices with a natural block 2 2 structure. An optimality condition leads to the problem of perturbing the (1,1) block of the matrix to achieve a speci®c inertia. We derive a perturbation of minimal norm, for any unitarily invariant norm, that increases the number of nonnegative eigenvalues by a given amount, and we s...
متن کاملSome Theorems on the Inertia of General Matrices
1.1. Much is known about the distribution of the roots of algebraic equations in half-planes. (Cf. the corresponding parts in the survey [l] by Marden.) In the case of matrix equations, however, there appears to be only one known general result concerning the location of the eigenvalues of a matrix in the left half-plane. This theorem is generally known as Lyapunov’s theorem: (L,) Let A be an n...
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ژورنال
عنوان ژورنال: Indiana University Mathematics Journal
سال: 2016
ISSN: 0022-2518
DOI: 10.1512/iumj.2016.65.5869