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Clément de Seguins Pazzis

  1. Sums of two triangularizable quadratic matrices over an arbitrary field.

    Tue, 09/13/2011 - 15:01 — SomNambul
    Authors: Clément de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    Let K be an arbitrary field, and a,b,c,d be elements of K such that the
    polynomials t^2-at-b and t^2-ct-d are split in K[t]. Given a square matrix M
    with entries in K, we give necessary and sufficient conditions for the
    existence of two matrices A and B such that M=A+B, A^2=a A+bI_n and B^2=c
    B+dI_n. Prior to this paper, such conditions were known in the case b=d=0, a<>0
    and c<>0, and also in the case a=b=c=d=0. Here, we complete the study, which
    essentially amounts to determining when a matrix is the sum of an idempotent
    and a square-zero matrix.

  2. The union of all orthogonal or symplectic groups.

    Fri, 08/27/2010 - 15:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    Given an endomorphism u of a finite-dimensional vector space (over an
    arbitrary field), we give necessary and sufficient conditions for the existence
    of a regular quadratic form (resp. a symplectic form) for which u is orthogonal
    (resp. symplectic). When the field of scalars has characteristic 2, we also
    give necessary and sufficient conditions for the existence of a regular
    symmetric bilinear form for which u is orthogonal. For the field of real
    numbers and for finite fields, we characterize the existence of a regular
    quadratic form in a given equivalence class for which u is orthogonal.

  3. The linear preservers of real diagonalizable matrices.

    Wed, 07/14/2010 - 15:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis, Bernard Rand&#xe9;
    Subjects: Rings and Algebras
    Abstract

    Using a recent result of Bogdanov and Guterman on the linear preservers of
    pairs of simultaneously diagonalizable matrices, we determine all the
    automorphisms of the vector space M_n(R) which stabilize the set of
    diagonalizable matrices. To do so, we investigate the structure of linear
    subspaces of diagonalizable matrices of M_n(R) with maximal dimension.

  4. The classification of large spaces of matrices with bounded rank.

    Mon, 04/05/2010 - 15:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    Given an arbitrary field K, let V be a linear subspace of M_n(K) consisting
    of matrices of rank lesser or equal to some r<n. A theorem of Atkinson and
    Lloyd states that, if dim V>nr-r+1 and #K>r, then either all the matrices of V
    vanish on some common (n-r)-dimensional subspace of K^n, or it is true of the
    matrices of its transpose V^t. Following some arguments of our recent proof of
    the Flanders theorem for an arbitrary field, we show that this result holds for
    any field.

  5. On product instability for large spaces of matrices.

    Mon, 04/05/2010 - 15:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    Let K denote a field. Given an arbitrary linear subspace V of M_n(K) of
    codimension lesser than n-1, a classical result states that V generates the
    K-algebra M_n(K). Here, we strengthen this in three ways: we show that M_n(K)
    is actually generated as a linear space by products of the form AB with A and B
    in V; we prove that every matrix in M_n(K) can be decomposed into a product of
    elements of V; finally, when V is a linear hyperplane of M_n(K) and n>2, we
    show that every matrix in M_n(K) is a product of two elements of V.

  6. The affine preservers of non-singular matrices.

    Wed, 03/31/2010 - 15:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    When K is an arbitrary field, we study the affine automorphisms of M_n(K)
    that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine
    subspaces of singular matrices, this is easily reduced to the known case of
    linear preservers when n>2 or #K>2. We include a short new proof of the more
    general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank.
    We also find that the group of affine transformations of M_2(F_2) that
    stabilize GL_2(F_2) does not consist solely of linear maps.

  7. The singular linear preservers of non-singular matrices.

    Wed, 02/10/2010 - 16:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    In this paper, we reduce the determination of the singular endomorphisms $f$
    of M_n(K) that stabilize GL_n(K) to the classification of n-dimensional
    division algebras over K. Our method, which is based upon Dieudonn\'e's theorem
    on singular subspaces of M_n(K), also yields a proof for the classical
    non-singular case.

  8. On small matrix subalgebras with a trivial centralizer.

    Mon, 01/25/2010 - 16:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    Given an integer n greater of equal to 3, we investigate the minimal
    dimension for a subalgebra of square matrices of order n with a trivial
    centralizer. It is shown that this dimension is 5 when n is even and 4 when it
    is odd. In this latter case, all 4-dimensional subalgebras with a trivial
    centralizer are explicitely computed.

  9. On the sum of the dimension of a matrix subalgebra and its centralizer.

    Fri, 12/25/2009 - 16:01 — SomNambul
    Authors: Cl&#xe9;ment de Seguins Pazzis
    Subjects: Rings and Algebras
    Abstract

    When $\mathbb{K}$ is a field, and $\mathcal{A}$ and $\mathcal{B}$ denote
    commuting subspaces of $\text{M}_n(\K)$ each of which contains a non-scalar
    matrix, we prove that $\dim \mathcal{A} +\dim \mathcal{B} \leq (n-1)^2+3$. We
    also give a complete description of the cases when equality holds.

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