查看更多>>摘要：Let F be an algebraically closed field and M-n be the n x n matrix algebra over F. A total graph of the full matrix algebra is the graph with M-n as vertices, and two distinct matrices A, B are adjacent if and only if A +B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of M-n preserving the singularity of matrix pencils A + lambda B. In particular, we characterize maps T-1, T-2: M-n & nbsp;->& nbsp;M-n satisfying the condition A + lambda B is singular if and only if T-1(A) + lambda T-2(B) is singular, for any A, B is an element of & nbsp;M-n and any non-zero lambda is an element of & nbsp;F. Namely, we prove that in this case T-1 = T-2 and they are of the form T-1(A) = T-2(A) = PAQ for all A is an element of & nbsp;M-n, or of the form T-1(A) = T-2(A) = PA(t)Q for all A is an element of & nbsp;M-n, where P, Q is an element of & nbsp;M-n are non-singular matrices. (C)& nbsp;2022 Elsevier Inc. All rights reserved.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional C*- algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. We illustrate how the procedure works for several examples relevant to quantum information theory. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：We characterize positive definiteness for some family of matrices. As an application, we derive the explicit value of the quadratic embedding constants of the path graphs.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
查看更多>>摘要：We study distinct (0, 1) matrices A and B, called Gram mates, such that AA(T) = BBT and A(T)A = (BB)-B-T. We characterize Gram mates where one can be obtained from the other by changing signs of some positive singular values. We classify Gram mates such that the rank of their difference is at most 2. Among such Gram mates, we further produce equivalent conditions in order that one is obtained from the other by changing signs of at most 2 positive singular values. Moreover, we provide some tools for constructing Gram mates where the rank of their difference is more than 2. Finally, we characterize non-isomorphic Gram mates whose difference is of rank 1 with some extra conditions.& nbsp;(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper we present a linear algorithm that estimates some physical parameters of a continuous-time system, described by an analytical mathematical model, when not all the state variables can be measured. The algorithm starts from the well-known subspace methods and applies some linear transformations to recover, at least partially, the estimated model in physical coordinates. Some analytical investigations and numerical experiments are shown for this method, which has general application within linear time -invariant (LTI) dynamical systems.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The tropical semiring R boolean OR {-infinity} is the semiring with addition "max" and multiplication "+". Tropical quadratic forms are represented by tropical symmetric matrices. Tropical quadratic forms with three variables define tropical projective quadratic plane curves. In this paper, we characterize tropical projective quadratic plane curves in terms of the eigenvalue problem for tropical matrices. In particular, we focus on the curved part of a tropical projective quadratic plane curve, that is, the cell never contained in any tropical projective line. We first prove that algebraic eigenvalues, i.e., roots of the characteristic polynomial, of a matrix express the minimum distance from the origin to the curved part. We then show that an algebraic eigenvector with respect to the minimum algebraic eigenvalue of a matrix indicates the direction to the nearest point in the curved part from the origin.(c) 2022 Elsevier Inc. All rights reserved.
Camenga, Kristin A.Rault, Patrick X.Spitkovsky, Ilya M.Yates, Rebekah B. Johnson...
27页
查看更多>>摘要：The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau-Wu number (i.e., the maximal number k(A) of orthonormal vectors xj such that the scalar products (Axj, xj) lie on the boundary of the numerical range of A) is computed for a class of arrowhead matrices A of arbitrary size, including dichotomous ones. These results are then used to completely classify all 4 x 4 matrices according to the values of their Gau-Wu numbers.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：Quadratic unitary Cayley graphs are a generalization of the well-known Paley graphs. Let Z(n) be the ring of integers modulo n. The quadratic unitary Cayley graph of Z(n), denoted by G(Zn), is the graph whose vertices are given by the elements of Z(n) and two vertices u, v is an element of Z(n) are adjacent if and only if u-v or v-u is a quadratic unit in Z(n). When p >= 3 is a prime and nu >= 1 is an integer, all the eigenvalues of G(Zp nu) have been given in [8]. In this paper, we improve the above result and obtain all the exact eigenvalues of G(Z2n) by a new approach. We also determine all the eigenvalues of G(Zn) for general n > 1. As an application, we characterize necessary and sufficient conditions on n such that G(Zn) is strongly regular. (c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The (k, r)-fan is the graph consisting of k copies of the complete graph Kr which intersect in a single vertex, and is denoted by Fk,r. Erdos et al. (1995) [14] determined the maximum number of edges in an n-vertex graph that does not contain F(k,3 )as a subgraph. Furthermore, Chen et al. (2003) [5] proved the analogous result on F-k,F-r for the general case r >= 3. In this paper, we show that for sufficiently large n, the graphs of order n that contain no copy of F-k,F-r and attain the maximum spectral radius are also edge-extremal. That is, such graphs must have ex(n, F-k,F-r) edges. (C)& nbsp;2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：For every m, n is an element of N and every field K, let M(m x n, K) be the vector space of the (m x n) matrices over K and let S(n, K) be the vector space of the symmetric (n x n)-matrices over K. We say that an affine subspace S of M(m x n, K) or of S(n, K) has constant rank r if every matrix of S has rank r. Define A(k)(m x n; r) = {S vertical bar S affine subspace of M(m x n, K) of constant rank r} A(sym)(k)(n; r) = {S vertical bar S affine subspace of S(n, K) of constant rank r} a(k)(m x n; r) = max{dim S vertical bar S is an element of A(k)(m x n, r)} a(sym)(k)(n; r) = max{dim S vertical bar S is an element of A(sym)(k)(n, r)}. In this paper we prove the following two formulas for r <= m <= n: a(sym)(R)(n; r) <= left perpendicularr/2right perpendicular (n - left perpendicularr/2right perpendicular) a(R)(m x n; r) = r(n - r) + r(r - 1)/2. (C) 2022 Elsevier Inc. All rights reserved.
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