coefficients of characteristic polynomial

jostpuur

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?

Stephen Tashi

Some Google searching turned up the "Leverrier-Faddeev" algorithm. One of the first hits was this PDF http://www.google.com/url?sa=t&sourc...mJIZXT55GHC5vg

I'm glad you asked this question. I'd never thought about it before.

jostpuur

Great!