On spectral theory and convexity
نویسندگان
چکیده
منابع مشابه
Convexity and Log Convexity for the Spectral Radius
The starting point of this paper is a theorem by J. F. C. Kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix. A related result of J. Cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements. The first section of this paper gives a new, unified pr...
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A spectral function on a formally real Jordan algebra is a real-valued function which depends only on the eigenvalues of its argument. One convenient way to create them is to start from a function f : R 7→ R which is symmetric in the components of its argument, and to define the function F (u) := f(λ(u)) where λ(u) is the vector of eigenvalues of u. In this paper, we show that this construction...
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We prove in this note the convexity of the functions u ◦ λ and more generally u ◦ λB on the space of Hermitian matrices, for B a fixed positive definite hermitian matrix, when u : R → R ∪ {+∞} is a symmetric convex function which is lower semi-continuous on R, and finite in at least one point of R. This is performed by using some optimisation techniques and a generalized Ky Fan inequality. To c...
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In recent years spectral clustering has become a standard method for data analysis used in a broad range of applications. In this paper we propose a new class of algorithms for multiway spectral clustering based on optimization of a certain class of functions over a sphere. These algorithms can be interpreted geometrically as recovering a discrete weighted simplex. The proposed methods have som...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1981
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-1981-0597867-6