Subadditivity of Eigenvalue Sums
نویسندگان
چکیده
Let f(t) be a nonnegative concave function on 0 ≤ t < ∞ with f(0) = 0, and letX,Y be n×nmatrices. Then it is known that ‖f(|X+Y |)‖1 ≤ ‖f(|X|)‖1+‖f(|Y |)‖1, where ‖ · ‖1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.
منابع مشابه
On eigenvalues and eigenvectors of subdirect sums
Some new properties of the eigenvalues of the subdirect sums are presented for the particular case of 1-subdirect sums. In particular, it is shown that if an eigenvalue λ is associated with certain blocks of matrix A or matrix B then λ is also an eigenvalue associated with the 1-subdirect sum A ⊕1 B. Some results concerning eigenvectors of the k-subdirect sum A⊕k B for an arbitrary positive int...
متن کاملOptimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices
This paper gives max characterizations for the sum of the largest eigen-values of a symmetric matrix. The elements which achieve the maximum provide a concise characterization of the generalized gradient of the eigenvalue sum in terms of a dual matrix. The dual matrix provides the information required to either verify rst-order optimality conditions at a point or to generate a descent direction...
متن کاملMathematical framework for detection and quantification of nonclassical correlations
Existing measures of bipartite nonclassical correlations that are typically characterized by nonvanishing nonlocalizable information under the zero-way CLOCC protocol are expensive in the computational cost. We define and evaluate economical measures on the basis of a new class of maps, eigenvalue-preserving-but-not-completelyeigenvalue-preserving (EnCE) maps. The class is in analogy to the cla...
متن کاملEigenvalue Statistics in Quantum Ideal Gases
The eigenvalue statistics of quantum ideal gases with single particle energies en = n α are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer α > 1 one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclas...
متن کاملNeumann Eigenvalue Sums on Triangles Are (mostly) Minimal for Equilaterals
We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n 3 . The result fails for n = 2 , because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenva...
متن کامل