Results 181 to 190 of about 5,647,728 (213)

A Measure of Asymmetry for Positive Semidefinite Matrices

Optimization, 2003
It is known that a continuous map is the gradient of a convex function if and only if it is cyclically monotone. Also, a differentiable map F is the gradient of a function if and only if the matrices F ′(x) are symmetric for all x in the domain. Based on this connection between symmetry and monotonicity, we define a measure of asymmetry for positive ...
Jean-Pierre Crouzeix
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Clarke Generalized Jacobian of the Projection Onto the Cone of Positive Semidefinite Matrices [PDF]

Set-Valued and Variational Analysis, 2006
International audienceThis paper studies the differentiability properties of the projection onto the cone of positive semidefinite matrices. In particular, the expression of the Clarke generalized Jacobian of the projection at any symmetric matrix is ...
Jérôme Malick, Hristo S Sendov
exaly   +2 more sources

Remarks on a Question of Bourin for Positive Semidefinite Matrices

Results in Mathematics, 2023
For positive semidefinite matrices \(A\) and \(B\) over the field of complex numbers, the authors prove the norm inequalities \[ |||A^tB^{1-t}+B^tA^{1-t}|||\leq 2^{2(t-3/4)}|||A+B||| , \] if \(t\in [3/4,1]\) and \[ |||A^tB^{1-t}+B^tA^{1-t}|||\leq 2^{2(1/4-t)}|||A+B||| , \] if \(t\in [0,1/4].\) Here, \(|||\cdot |||\) is any unitarily invariant norm.
Hayajneh, Mostafa, Hayajneh, Saja, Kittaneh, Fuad   +2 more
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Semidefinite Programming in the Space of Partial Positive Semidefinite Matrices

SIAM Journal on Optimization, 2003
We build upon the work of Fukuda et al. [SIAM J. Optim., 11 (2001), pp. 647--674] and Nakata et al. [Math. Program., 95 (2003), pp. 303--327], in which the theory of partial positive semidefinite matrices was applied to the semidefinite programming (SDP) problem as a technique for exploiting sparsity in the data.
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Positive Semidefinite Matrices

2018
Positive semidefinite (PSD) and positive definite (PD) matrices are closely connected with Euclidean distance matrices. Accordingly, they play a central role in this monograph. This chapter reviews some of the basic results concerning these matrices.
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Fractional Hadamard Powers of Positive Semidefinite Matrices

Linear Algebra and Its Applications, 2003
The authors consider the class \(\varphi_n\) of all real positive semidefinite \(n\times n\) matrices, and the subclass \(\varphi^+_n\) of all \(A\in\varphi_n\) with non-negative entries. For a positive, non-integer number \(\alpha\) and some \(A\in \varphi_n^+\), when will the fractional Hadamard power \(A^{\diamondsuit \alpha}\) again belong to ...
Fischer, P., Stegeman, J.D.
exaly   +4 more sources

Positive Semidefinite Matrices

1999
This chapter studies the positive semidefinite matrices, concentrating primarily on the inequalities of this type of matrix. The main goal is to present the fundamental results and show some often-used techniques. Section 7.1 gives the basic properties, Section 7.2 treats the L¨owner partial ordering of positive semidefinite matrices, and Section 7.3 ...
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Norm inequalities for positive semidefinite matrices

Wuhan University Journal of Natural Sciences, 2012
This paper aims to discuss some inequalities involving unitarily invariant norms and positive semidefinite matrices. By using properties of unitarily invariant norms, we obtain two inequities involving unitarily invariant norms and positive semidefinite matrices, which generalize the result obtained by Bhatia and Kittaneh.
Limin Zou, Yanqiu Wu
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A new Farkas lemma for positive semidefinite matrices

IEEE Transactions on Automatic Control, 1995
Let \(A\) be a linear mapping of the vector space of symmetric real matrices of a given size. Let \(S\) be a closed convex cone of positive matrices. The Farkas lemma characterizes the consistency of the system \(Ax= b\), \(x\in S\). The author obtains a variant that avoids the closeness assumption imposed on \(S+ \ker A\).
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Estimation of Positive Semidefinite Correlation Matrices by Using Convex Quadratic Semidefinite Programming

Neural Computation, 2009
The correlation matrix is a fundamental statistic that used in many fields. For example, GroupLens, a collaborative filtering system, uses the correlation between users for predictive purposes. Since the correlation is a natural similarity measure between users, the correlation matrix may be used as the Gram matrix in kernel methods.
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