Browsing by Subject "commuting matrices"

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    Commutative cocycles and stable bundles over surfaces
    (De Gruyter, 2019-11) Ramras, Daniel A.; Villareal, Bernardo; Mathematical Sciences, School of Science
    Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, Gómez, Gritschacher, Lind and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O(2)-valued cocycles and are analyzed via the point-wise inversion operation on commutative cocycles.
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    Constructing invariant subspaces as kernels of commuting matrices
    (Elsevier, 2019-12) Cowen, Carl C.; Johnston, William; Wahl, Rebecca G.; Mathematical Sciences, School of Science
    Given an n n matrix A over C and an invariant subspace N, a straightforward formula constructs an n n matrix N that commutes with A and has N = kerN. For Q a matrix putting A into Jordan canonical form, J = Q􀀀1AQ, we get N = Q􀀀1M where M= ker(M) is an invariant subspace for J with M commuting with J. In the formula J = PZT􀀀1Pt, the matrices Z and T are m m and P is an n m row selection matrix. If N is a marked subspace, m = n and Z is an n n block diagonal matrix, and if N is not a marked subspace, then m > n and Z is an m m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a nite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A.