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Sang-il Oum - One of the best experts on this subject based on the ideXlab platform.
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Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices
Linear Algebra and its Applications, 2012Co-Authors: Sang-il OumAbstract:Abstract We prove that every infinite sequence of skew-symmetric or symmetric matrices M 1 , M 2 , … over a fixed finite field must have a pair M i , M j ( i j ) such that M i is isomorphic to a Principal Submatrix of the Schur complement of a nonsingular Principal Submatrix in M j , if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour’s theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle’s theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum’s theorem for graphs of bounded rank-width with respect to pivot-minors.
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Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices (extended abstract)
Electronic Notes in Discrete Mathematics, 2011Co-Authors: Sang-il OumAbstract:Abstract We prove that every infinite sequence of skew-symmetric or symmetric matrices M 1 , M 2 , … over a fixed finite field must have a pair M i , M j ( i j ) such that M i is isomorphic to a Principal Submatrix of the Schur complement of a nonsingular Principal Submatrix in M j , if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymourʼs theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittleʼs theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oumʼs theorem for graphs of bounded rank-width with respect to pivot-minors.
Lei Zhang - One of the best experts on this subject based on the ideXlab platform.
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Inverse eigenvalue problems for bisymmetric matrices under a central Principal Submatrix constraint
Linear and Multilinear Algebra, 2011Co-Authors: Lijun Zhao, Lei ZhangAbstract:This article considers an inverse eigenvalue problem for bisymmetric matrices under a central Principal Submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of bisymmetric matrices and their central Principal submatrices. Then we study a special form for the matrix of independent eigenvectors for a bisymmetric matrix. Based on these, we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.
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least squares solutions to ax b for bisymmetric matrices under a central Principal Submatrix constraint and the optimal approximation
Linear Algebra and its Applications, 2008Co-Authors: Lijun Zhao, Lei ZhangAbstract:Abstract A matrix A ∈ R n × n is called a bisymmetric matrix if its elements a i , j satisfy the properties a i , j = a j , i and a i , j = a n - j + 1 , n - i + 1 for 1 ⩽ i , j ⩽ n . This paper considers least squares solutions to the matrix equation AX = B for A under a central Principal Submatrix constraint and the optimal approximation. A central Principal Submatrix is a Submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central Principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.
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Least squares solutions to AX = B for bisymmetric matrices under a central Principal Submatrix constraint and the optimal approximation
Linear Algebra and its Applications, 2008Co-Authors: Lijun Zhao, Lei ZhangAbstract:Abstract A matrix A ∈ R n × n is called a bisymmetric matrix if its elements a i , j satisfy the properties a i , j = a j , i and a i , j = a n - j + 1 , n - i + 1 for 1 ⩽ i , j ⩽ n . This paper considers least squares solutions to the matrix equation AX = B for A under a central Principal Submatrix constraint and the optimal approximation. A central Principal Submatrix is a Submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central Principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.
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a kind of inverse eigenvalue problems of jacobi matrix
Applied Mathematics and Computation, 2006Co-Authors: Juan Peng, Xiyan Hu, Lei ZhangAbstract:This paper considers the problem of constructing two Jacobi matrices from their Principal Submatrix and specially ordered defective eigenpairs. The two Jacobi matrices are the same except for the entries in the last column and last row. The necessary and sufficient condition of solvability is derived. Furthermore, two algorithms and numerical examples are given.
S. V. Savchenko - One of the best experts on this subject based on the ideXlab platform.
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On the change of the Jordan form under the transition from the adjacency matrix of a vertex-transitive digraph to its Principal Submatrix of co-order one
Linear Algebra and its Applications, 2005Co-Authors: S. V. SavchenkoAbstract:Abstract Let J ( λ ; n 1 , …, n k ) be the set of matrices A such that λ is an eigenvalue of A and n 1 ⩽ ⋯ ⩽ n k are the sizes of the Jordan blocks associated with λ . For a given index v of A , denote by A − v the Principal Submatrix of co-order one obtained from A by deleting the v th row and column. In the present paper, all possible changes of the part of the Jordan form corresponding to λ under the transition from A to A − v are determined for matrices A ∈ J ( λ ; n 1 , …, n k ) such that for the eigenvalue λ of both A and A ⊤ , there exists a Jordan chain of the largest length n k whose eigenvector has nonzero v th entry. In particular, it is shown that for almost every matrix A ∈ J ( λ ; n 1 , …, n k ), n 1 , …, n k −1 are the sizes of Jordan blocks for λ considered as an eigenvalue of A − v . Moreover, it is also proved that if A is the adjacency matrix of a vertex-transitive digraph and k ⩾ 2, then the change n 1 , …, n k → n 1 , …, n k −2 , 2 n k −1 − 1 holds for the eigenvalue λ under the transition from A to A − v . In the case of k = 1, λ is a simple eigenvalue of A and does not belong to the spectrum of A − v .
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On the Perron roots of Principal submatrices of co-order one of irreducible nonnegative matrices
Linear Algebra and its Applications, 2003Co-Authors: S. V. SavchenkoAbstract:Let A be an irreducible nonnegative matrix and λ(A) be the Perron root (spectral radius) of A. Denote by λ min (A) the minimum of the Perron roots of all the Principal submatrices of co-order one. It is well known that the interval (λ min (A), λ(A)) does not contain any eigenvalues of A. Consider any Principal Submatrix A - v of co-order one whose Perron root is equal to λ min (A). We show that the Jordan structure of λ min (A) as an eigenvalue of A is obtained from that of the Perron root of A - v as follows: one largest Jordan block disappears and the others remain the same. So, if only one Jordan block corresponds to the Perron root of the Submatrix, then λ min (A) is not an eigenvalue of A. By Schneider's theorem, this holds if and only if there is a Hamiltonian chain in the singular digraph of A - v. In the general case the Jordan structure for the Perron root of the Submatrix A - v and therefore that for the eigenvalue λ min (A) of A can be arbitrary. But if the Perron root λ(A - w) of a Principal Submatrix A - w of co-order one is strictly greater than λ min (A), then λ(A - w) is a simple eigenvalue of A - w. We also obtain different representations for the generalized eigenvectors corresponding to the eigenvalues of A contained in the annulus {λ: λ min (A) < |λ| < λ(A)}.
Charles R. Johnson - One of the best experts on this subject based on the ideXlab platform.
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Spectra of Tridiagonal Matrices over a Field.
arXiv: Classical Analysis and ODEs, 2018Co-Authors: Roberto S. Costas-santos, Charles R. JohnsonAbstract:We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing Principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and sufficient that the necessarily real eigenvalues are distinct and those of the Principal Submatrix strictly interlace. So this case is very restrictive. By contrast, for a general field, the requirements on the two spectra are much less restrictive. In particular, in the real or complex case, the $n$-by-$n$ characteristic polynomial is arbitrary (so that the algebraic multiplicities may be anything in place of all 1's in the classical cases) and that of the Principal Submatrix is the complement of a lower dimensional algebraic set (and so relatively free). Explicit conditions are given.
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The change in eigenvalue multiplicity associated with perturbation of a diagonal entry
Linear and Multilinear Algebra, 2012Co-Authors: Charles R. Johnson, António Leal-duarte, Carlos M. SaiagoAbstract:We investigate the relation between perturbing the i-th diagonal entry of A ∈ ℳ n (𝔽) and extracting the Principal Submatrix A(i) from A with respect to the possible changes in multiplicity of a given eigenvalue. A complete description is given and used to both generalize and improve prior work about Hermitian matrices whose graph is a given tree.
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The change in eigenvalue multiplicity associated with perturbation of a diagonal entry of the matrix
2009Co-Authors: Charles R. Johnson, António Leal Duarte, Carlos M. SaiagoAbstract:Here we investigate the relation between perturbing the i-th diagonal entry of A ∈ Mn(F) and extracting the Principal Submatrix A(i) from A with respect to the possible changes in multiplicity of a given eigenvalue. A complete description is given and used to both generalize and improve prior work about Hermitian matrices whose graph is a given tree.
Ana M. Urbano - One of the best experts on this subject based on the ideXlab platform.
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All Jordan canonical forms of irreducible totally non-negative matrices
Linear and Multilinear Algebra, 2019Co-Authors: Begoña Cantó, Rafael Cantó, Ana M. UrbanoAbstract:Let A∈Rn×n be an irreducible totally non-negative matrix with rank r and Principal rank p, that is, every minor of A is non-negative and p is the size of the largest invertible Principal Submatrix ...
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On the maximum rank of totally nonnegative matrices
Linear Algebra and its Applications, 2018Co-Authors: Rafael Cantó, Ana M. UrbanoAbstract:Abstract Let A ∈ R n × n be a totally nonnegative matrix with Principal rank p, that is, every minor of A is nonnegative and p is the size of the largest invertible Principal Submatrix of A. We introduce the sequence of the first p-indices of A as the first initial row and column indices of a p × p invertible Principal Submatrix of A with rank p. Then, we study the linear dependence relations between the rows and columns indexed by the sequence of the first p-indices of A and the remaining of its rows and columns. These relations, together with the irreducibility property of some submatrices of A, allow us to present an algorithm that calculates the maximum rank of A as a function of the distribution of the first p-indices. Finally, we present a method to construct n × n totally nonnegative matrices with given rank r, Principal rank p and a specific sequence of the first p-indices.