Sufficient conditions for symmetric matrices to have exactly one positive eigenvalue
Special Matrices, 2020 Let A = [aij] be a real symmetric matrix. If f : (0, ∞) → [0, ∞) is a Bernstein function, a sufficient condition for the matrix [f (aij)] to have only one positive eigenvalue is presented.
Al-Saafin Doaa, Garloff Jürgen
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The inertia of the symmetric approximation for low-rank matrices [PDF]
, 2017 © 2017 Informa UK Limited, trading as Taylor & Francis Group In many areas of applied linear algebra, it is necessary to work with matrix approximations.
Casanellas Rius, Marta +2 more
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Special Matrices, 2017 We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue ...
Toyonaga Kenji, Johnson Charles R.
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Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions [PDF]
, 2017 Fractional minimum positive semidefinite rank is defined from r-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement.
Hogben, L +3 more
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A constructive arbitrary-degree Kronecker product decomposition of tensors [PDF]
, 2016 We propose the tensor Kronecker product singular value decomposition~(TKPSVD) that decomposes a real $k$-way tensor $\mathcal{A}$ into a linear combination of tensor Kronecker products with an arbitrary number of $d$ factors $\mathcal{A} = \sum_{j=1}^R ...
Batselier, Kim, Wong, Ngai
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A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric [PDF]
, 2017 We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the existing optimization-based approach, we work from a purely matrix-algebraic perspective.
Zimmermann, Ralf
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Concrete minimal 3 × 3 Hermitian matrices and some general cases
Demonstratio Mathematica, 2017 Given a Hermitian matrix M ∈ M3(ℂ) we describe explicitly the real diagonal matrices DM such that ║M + DM║ ≤ ║M + D║ for all real diagonal matrices D ∈ M3(ℂ), where ║ · ║ denotes the operator norm.
Klobouk Abel H., Varela Alejandro
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Nonnegative definite hermitian matrices with increasing principal minors
, 2013 A nonzero nonnegative definite hermitian m by m matrix A has increasing principal minors if the value of each principle minor of A is not less than the value each of its subminors. For $m>1$ we show $A$ has increasing principal minors if and only if $A^{-
Friedland, Shmuel
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On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs
, 2015 The principal permanent rank characteristic sequence is a binary sequence $r_0 r_1 \ldots r_n$ where $r_k = 1$ if there exists a principal square submatrix of size $k$ with nonzero permanent and $r_k = 0$ otherwise, and $r_0 = 1$ if there is a zero ...
Horn, Paul +5 more
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Linear preservers and quantum information science
, 2012 Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$. Suppose $|\cdot|$ is the Ky Fan $k$-norm with $1 \le k \le mn$, or
Fosner, Ajda +3 more
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