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Vladimir V. Sergeichuk - One of the best experts on this subject based on the ideXlab platform.

  • each n by n Matrix with n 1 is a sum of 5 coninvolutory matrices
    Linear Algebra and its Applications, 2016
    Co-Authors: Ma Nerissa M Abara, Vladimir V. Sergeichuk, Dennis I. Merino, Viacheslav I Rabanovich, John Patrick Sta Maria
    Abstract:

    Abstract An n × n Complex Matrix A is called coninvolutory if A ¯ A = I n and skew-coninvolutory if A ¯ A = − I n (which implies that n is even). We prove that each Matrix of size n × n with n > 1 is a sum of 5 coninvolutory matrices and each Matrix of size 2 m × 2 m is a sum of 5 skew-coninvolutory matrices. We also prove that each Square Complex Matrix is a sum of a coninvolutory Matrix and a condiagonalizable Matrix. A Matrix M is called condiagonalizable if M = S ¯ − 1 D S in which S is nonsingular and D is diagonal.

  • Topological classification of sesquilinear forms: Reduction to the nonsingular case
    Linear Algebra and its Applications, 2016
    Co-Authors: Carlos M. Da Fonseca, Tetiana Rybalkina, Vladimir V. Sergeichuk
    Abstract:

    Abstract Two sesquilinear forms Φ : C m × C m → C and Ψ : C n × C n → C are called topologically equivalent if there exists a homeomorphism φ : C m → C n (i.e., a continuous bijection whose inverse is also a continuous bijection) such that Φ ( x , y ) = Ψ ( φ ( x ) , φ ( y ) ) for all x , y ∈ C m . R.A. Horn and V.V. Sergeichuk in 2006 constructed a regularizing decomposition of a Square Complex Matrix A ; that is, a direct sum S A S ⁎ = R ⊕ J n 1 ⊕ ⋯ ⊕ J n p , in which S and R are nonsingular and each J n i is the n i -by- n i singular Jordan block. In this paper, we prove that Φ and Ψ are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands J n i and replacement of R ∈ C r × r by a nonsingular Matrix R ′ ∈ C r × r such that R and R ′ are the matrices of topologically equivalent forms C r × C r → C . Analogous results for bilinear forms over C and over R are also obtained.

  • Miniversal deformations of matrices under *congruence and reducing transformations
    Linear Algebra and its Applications, 2014
    Co-Authors: Andrii Dmytryshyn, Vyacheslav Futorny, Vladimir V. Sergeichuk
    Abstract:

    Arnold (1971) [1] constructed a miniversal deformation of a Square Complex Matrix under similarity; that is, a simple normal form to which not only a given Square Matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of Square Complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].

  • An informal introduction to perturbations of matrices determined up to similarity or congruence
    The São Paulo Journal of Mathematical Sciences, 2014
    Co-Authors: Lena Klimenko, Vladimir V. Sergeichuk
    Abstract:

    The reductions of a Square Complex Matrix A to its canon- ical forms under transformations of similarity, congruence, or *con- gruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the entries of A. We sur- vey results about their behavior under perturbations of A and about normal forms of all matrices A + E in a neighborhood of A with re- spect to similarity, congruence, or *congruence. These normal forms are called miniversal deformations of A; they are not uniquely deter- mined by A + E, but they are simple and depend continuously on the entries of E.

  • A criterion for unitary similarity of upper triangular matrices in general position
    Linear Algebra and its Applications, 2011
    Co-Authors: Douglas Farenick, Vladimir V. Sergeichuk, Vyacheslav Futorny, Tatiana G. Gerasimova, Nadya Shvai
    Abstract:

    Abstract Each Square Complex Matrix is unitarily similar to an upper triangular Matrix with diagonal entries in any prescribed order. Let A = [ a ij ] and B = [ b ij ] be upper triangular n × n matrices that • are not similar to direct sums of Square matrices of smaller sizes, or • are in general position and have the same main diagonal. We prove that A and B are unitarily similar if and only if ‖ h ( A k ) ‖ = ‖ h ( B k ) ‖ for all h ∈ C [ x ] and k = 1 , … , n , where A k : = [ a ij ] i , j = 1 k and B k : = [ b ij ] i , j = 1 k are the leading principal k × k submatrices of A and B, and ‖ · ‖ is the Frobenius norm.

M.j. Tsatsomeros - One of the best experts on this subject based on the ideXlab platform.

  • The envelope of tridiagonal Toeplitz matrices and block-shift matrices
    Linear Algebra and its Applications, 2017
    Co-Authors: Aik. Aretaki, Panayiotis Psarrakos, M.j. Tsatsomeros
    Abstract:

    Abstract The envelope of a Square Complex Matrix is a spectrum encompassing region in the Complex plane. It is contained in and is akin to the numerical range in the sense that the envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curves, rather than half-planes. In this article, the geometry and properties of the envelopes of special matrices are examined. In particular, symmetries of the envelope of a tridiagonal Toeplitz Matrix are obtained, and the envelopes of block-shift matrices, Jordan blocks and 2 × 2 matrices are explicitly characterized.

  • Extremal properties of ray-nonsingular matrices
    Discrete Mathematics, 2000
    Co-Authors: G.y. Lee, J.j. Mcdonald, B.l. Shader, M.j. Tsatsomeros
    Abstract:

    A ray-nonsingular Matrix is a Square Complex Matrix, A, such that each Complex Matrix whose entries have the same arguments as the corresponding entries of A, is nonsingular. Extremal properties of ray-nonsingular matrices are studied in this paper. Combinatorial and probabilistic arguments are used to prove that if the order of a ray-nonsingular Matrix is at least 6, then it must contain a zero entry, and that if each of its rows and columns have an equal number, k, of nonzeros, then k613. c 2000 Elsevier Science B.V. All rights reserved.

  • Extremal properties of ray-nonsingular matrices
    Discrete Mathematics, 2000
    Co-Authors: G.y. Lee, J.j. Mcdonald, B.l. Shader, M.j. Tsatsomeros
    Abstract:

    AbstractA ray-nonsingular Matrix is a Square Complex Matrix, A, such that each Complex Matrix whose entries have the same arguments as the corresponding entries of A, is nonsingular. Extremal properties of ray-nonsingular matrices are studied in this paper. Combinatorial and probabilistic arguments are used to prove that if the order of a ray-nonsingular Matrix is at least 6, then it must contain a zero entry, and that if each of its rows and columns have an equal number, k, of nonzeros, then k⩽13

Fazlollah Soleymani - One of the best experts on this subject based on the ideXlab platform.

F. Khaksar Haghani - One of the best experts on this subject based on the ideXlab platform.

Kh D Ikramov - One of the best experts on this subject based on the ideXlab platform.

  • An Attempt of Spectral Theory for ∗-Congruence Transformations
    Journal of Mathematical Sciences, 2020
    Co-Authors: Kh D Ikramov
    Abstract:

    The paper discusses the possibility of reducing a Square Complex Matrix A to a direct sum of smaller matrices by using ∗-congruence transformations. It turns out that this possibility is related to appropriate partitions of the spectrum of the coSquare of A. This makes it possible to associate the direct summands of the sum with subsets of the latter spectrum.

  • A Rational Criterion for Congruence of Square Matrices
    Journal of Mathematical Sciences, 2019
    Co-Authors: Kh D Ikramov
    Abstract:

    With a Square Complex Matrix A the Matrix pair consisting of its symmetric S ( A ) = ( A + A ^ T )/2 and skew-symmetric K ( A ) = ( A − A ^ T )/2 parts is associated. It is shown that Square matrices A and B are congruent if and only if the associated pairs ( S ( A ), K ( A )) and ( S ( B ), K ( B )) are (strictly) equivalent. This criterion can be verified by a rational calculation, provided that the entries of A and B are rational or rational Gaussian numbers.

  • Binormal Matrices
    Journal of Mathematical Sciences, 2018
    Co-Authors: Kh D Ikramov
    Abstract:

    A Square Complex Matrix A is said to be binormal if the associated matrices A^*A and AA^* commute. This Matrix class yields a meaningful finite-dimensional extension of the concept of normality. The paper can be regarded as a survey of properties of binormal matrices.

  • Effective algorithms for deComplexifying a Matrix by unitary similarities or congruences
    Mathematical Notes, 2012
    Co-Authors: Kh D Ikramov
    Abstract:

    It is required to verify whether a given Complex n × n Matrix A can be made real by a similarity or a congruence transformation. Algorithms for solving these two problems are proposed and justified under the additional assumption that A is irreducible in the former case and $$A_L = \bar AA$$ is irreducible in the latter case. The irreducibility of a Square Complex Matrix means that no unitary similarity transformation converts this Matrix into a direct sum of smaller matrices. The proposed algorithms are effective in the sense that their implementation requires a finite number of arithmetic operations.

  • a note on an unusual type of polar decomposition
    Linear Algebra and its Applications, 2008
    Co-Authors: Heike Fasbender, Kh D Ikramov
    Abstract:

    Motivated by applications in the theory of unitary congruence, we introduce the factorization of a Square Complex Matrix A of the form A=SU, where S is Complex symmetric and U is unitary. We call this factorization a symmetric–unitary polar decomposition or an SUPD. It is shown that an SUPD exists for every Matrix A and is always nonunique. Even the symmetric factor S can be chosen in infinitely many ways. Nevertheless, we show that many properties of the conventional polar decomposition related to normal matrices have their counterparts for the SUPD, provided that normal matrices are replaced with conjugate–normal ones.