- #1
jostpuur
- 2,112
- 19
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
<br /> \det (A - \lambda) = a_n\lambda^n + \cdots + a_1\lambda + a_0,\quad\quad\forall\lambda\in\mathbb{C}<br />
If n=2,
<br /> a_2 = 1,\quad a_1 = -\textrm{tr}(A),\quad a_0 = \textrm{det}(A).<br />
If n=3,
<br /> a_3 = -1,\quad a_2 = \textrm{tr}(A),\quad a_0 = \textrm{det}(A)<br />
and
<br /> 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}<br />
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?
<br /> \det (A - \lambda) = a_n\lambda^n + \cdots + a_1\lambda + a_0,\quad\quad\forall\lambda\in\mathbb{C}<br />
If n=2,
<br /> a_2 = 1,\quad a_1 = -\textrm{tr}(A),\quad a_0 = \textrm{det}(A).<br />
If n=3,
<br /> a_3 = -1,\quad a_2 = \textrm{tr}(A),\quad a_0 = \textrm{det}(A)<br />
and
<br /> 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}<br />
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?