Coefficients of characteristic polynomial

[jostpuur]

I want to write an algorithm that gives as output the numbers $a_n,\ldots, a_1,a_0$, when a matrix $A\in\mathbb{R}^{n\times n}$ is given as input, such that

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

If $n=2$,

$$a_2 = 1,\quad a_1 = -\textrm{tr}(A),\quad a_0 = \textrm{det}(A).$$

If $n=3$,

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

So the coefficients $a_n,a_{n-1},a_0$ are easy, but $a_{n-2},\ldots, a_1$ 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&sour...hd3wCQ&usg=AFQjCNHYULZI0U-iyTFmmJIZXT55GHC5vg

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

 

[jostpuur]

"Leverrier-Faddeev" algorithm

http://www.google.com/url?sa=t&sour...hd3wCQ&usg=AFQjCNHYULZI0U-iyTFmmJIZXT55GHC5vg

Great!