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Upper Hessenberg zero-column sum matrix

  1. May 2, 2012 #1
    Let A be an (n x n) upper Hessenberg matrix with the following constraints:
    a_{j,j} < 0 , j=1,...,n
    a_{j,j}= - sum a_{i,j} [i≠j] , j=1,...,n-1
    a_{n,n} > - sum a_{i,n} [i≠n]

    -1 2 0 2 0
    1 -5 4 0 0
    0 3 -9 1 0
    0 0 5 -5 3
    0 0 0 2 -7

    I want to prove A is diagonalizable (I am pretty sure it is, but haven't
    found a formal demostration)

    Hint: If we make a_{1,4}=0; a_{4,4}=-3, then it is a tridiagonal matrix
    similar to a Hermitian one, and therefore diagonalizable.

    Thanks for your help,
  2. jcsd
  3. May 2, 2012 #2

    I know what is the Heisenberg group of upper 3 x 3 matrices, I know what is an upper matrix and I even

    have some idea what is Heisenberg's Matrix mechanics...

    Your matrix matrix is neither 3 x 3, nor upper and, as far as I know, now Heisenberg's as, even if we'd talk of higher dimensions

    Heisenberg groups, we need upper matrices, so: what exactly is your matrix, anyway??

    Anyway, using this nice site http://calculator-online.org/s/matrix/sobstvennyie/ it seems to be your matrix has

    5 different complex eigenvalues so it is diagonalizable over the complex field.

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