查看更多>>摘要：We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions. For small matrices, we provide equations; for larger matrices, we explain how to use representation theory to find equations. We describe the ring of invariants under the action of the orthogonal group. For the subvariety of diagonal matrices, we give the degree. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper we give a Specht-type characterization for unitary equivalence of operator tuples in the Cowen-Douglas class via their point-wise localizations. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper, we prove several unitarily invariant norm inequalities for positive semidefinite matrices. Some of these results give generalizations of earlier known inequalities. Among other applications of our inequalities, we obtain the commutator inequality parallel to XY - YX parallel to <= parallel to X parallel to parallel to Y parallel to + 1/2 parallel to X*Y - YX*parallel to for all n x ncomplex matrices X, Y. Here, parallel to.parallel to denotes the spectral norm. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：We show that any m x m matrix Mwith integer entries and det M = Delta not equal 0 can be equipped by a finite digit set D subset of Z(m) such that any integer m-dimensional vector belongs to the set Fin(D)(M) = {Sigma(k epsilon I) M(k)d(k) : theta not equal I finite subsetof Z and d(k) epsilon D for each k epsilon I}subset of boolean OR(k epsilon N) 1/Delta(k) Z(m). We also characterize the matrices M for which the sets Fin(D)(M) and boolean OR(k epsilon N) 1/Delta(k) Z(m) coincide. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：We investigate distribution of eigenvalues of growing size Toeplitz matrices [a(n+k-j)](1 <= j,k <= n) as n -> infinity, when the entries {a(j)} are "smooth" in the sense, for example, that for some alpha > 0, a(j-1)a(j+1)/a(j)(2) = 1 - 1/a(j) (1 + o(1)), j -> infinity. Typically they are Maclaurin series coefficients of an entire function. We establish that when suitably scaled, the eigenvalue counting measures have limiting support on [0,1], and under mild additional smoothness conditions, the universal scaled and weighted limit distribution is vertical bar pi logt vertical bar(-1/2) dt on [0,1]. (C) 2021 Elsevier Inc. All rights reserved.
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