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Coefficients of characteristic polynomial

  1. Sep 5, 2011 #1
    I want to write an algorithm that gives as output the numbers [itex]a_n,\ldots, a_1,a_0[/itex], when a matrix [itex]A\in\mathbb{R}^{n\times n}[/itex] is given as input, such that

    [tex]
    \det (A - \lambda) = a_n\lambda^n + \cdots + a_1\lambda + a_0,\quad\quad\forall\lambda\in\mathbb{C}
    [/tex]

    If [itex]n=2[/itex],

    [tex]
    a_2 = 1,\quad a_1 = -\textrm{tr}(A),\quad a_0 = \textrm{det}(A).
    [/tex]

    If [itex]n=3[/itex],

    [tex]
    a_3 = -1,\quad a_2 = \textrm{tr}(A),\quad a_0 = \textrm{det}(A)
    [/tex]
    and
    [tex]
    a_1 = -A_{11}A_{22} - A_{22}A_{33} - A_{33}A_{11} + A_{12}A_{21} + A_{23}A_{32} + A_{31}A_{13}
    [/tex]

    So the coefficients [itex]a_n,a_{n-1},a_0[/itex] are easy, but [itex]a_{n-2},\ldots, a_1[/itex] get difficult. Is there any recursion formula for them?
     
  2. jcsd
  3. Sep 6, 2011 #2

    Stephen Tashi

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    Science Advisor

  4. Sep 7, 2011 #3
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