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Homework Help: Help Showing d/dt of Char. Polynomial

  1. Feb 14, 2005 #1
    I've been working on a problem all day but I can't seem to go anywhere with it. I have to prove the following equation holds, in which [tex]P_A(t)[/tex] refers to the characteristic polynomial of an n by n matrix A and is expressed in terms of t. Also, [tex]P_A_i(t)[/tex] is the characteristic polynomial of the principal submatrix formed by removing row i and column i from A, also expressed in terms of t. The equation is:

    [tex]\frac{d}{dt}P_A(t) = \sum_{i=1}^n P_A_i(t) [/tex]

    I've been trying to prove it by induction, using the following formula for the left side. In the formula, for a given k, [tex]E_k(A)[/tex] is the sum of the k-by-k principal minors of A.:

    [tex]\frac{d}{dt}P_A(t) = nt^n^-^1 - (n-1)E_1(A)t^n^-^2 + (n-2)E_2(A)t^n - ... \pm E_n_-_1(A) [/tex]

    I'm not sure how to arrive at the right side of the first equation at all. I've tried rearranging it, but I don't know how to make up for the fact that all of the (n-1) by (n-1) principal submatrices of A are different. My only interpretation of the right hand side is as the sum of the characteristic polynomials of all (n-1) by (n-1) principal submatrices of A, but I can't seem to draw anything from that. Am I on an entirely wrong path, or am I just missing a step somewhere?

    Thanks,
    Oobob
     
    Last edited: Feb 14, 2005
  2. jcsd
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