u can use Horner's scheme (are you familiar with it) ? sorry I forgot do they acquainted you with that on your level of education?
or u can :
so u have
I need to learn to find eigenvalues and vectors.
I can find the Characteristic polynomial (and if i have the eigenvalues i can find the vectors)
However i cannot get from the Characteristic polynomial to the eigenvalues
for example if i have a 3x3 matrix
(1 0 4)
(0 2 0)
(3 1 -3)
The Characteristic polynomial is x^3 - 19x + 30 = 0 (x=lambda)
I know that the eigenvalues are Real eigenvalues:
{-5, 2, 3}
and then i can work out the eigenvectors.
But how do you get from this step (x^3 - 19x + 30 = 0) to the eigenvalues.
Thankyou!
Another thing you can do is use the "rational roots theorem": if is a polynomial equation with integer coefficients and is a rational root of the equation, then the denominator, n, must divide the leading coefficient, , and the numerator, m must divide the constant term, .
Here, your equation is , a polynomial equation with integer coefficients. The leading coefficient is 1 and the only integers that divide 1 are 1 and -1. Since the denominator of any rational root must be 1 or -1, the only rational roots must be integers. The constant term is 30 so any rational root must be a factor of 30. Since 30= 2(3)(5), the only possible rational roots are 1, -1, 2, -2, 3, -3, 5, -5, 6, -6, 10, -10, 15, -15, 30, and -30. Just put each of those into the equation to see whether or not they satisfy the equation and you will see that 2, 3, and -5 satisfy it.
Note- this does not guarentee that there are any rational roots to a polynomial equation but if there are, it will find them.
We should point out that, in general, finding eigenvalues is not at all an easy task!