# Eigen values and cubic roots question

1. Dec 2, 2009

### iamsmooth

So I found the characteristic equation of a matrix, and I know the roots of the equation are supposed to be the eigenvalues. However, my equation is:

$$\lambda^3-2\lambda^2$$

I have double checked different row expansions to make sure this answer is correct. So don't worry about how I came to get that equation.
I'm just not sure how to get roots from this. Would it be:

$$\lambda^2(\lambda-2)$$

So that the roots are 0 and 2?

Basically I have trouble with cubic roots, I guess this is less of a question about eigenvalues than it is about cubic roots.

Thanks.

2. Dec 2, 2009

### CompuChip

You are right about your example (note 0 is a double root).
In general cubic equations are no fun. Your options are:
1. Factor, like you did above
2. Guess an answer $\lambda_0$, then divide out $(\lambda-\lambda_0)$ and solve the resulting quadratic equation. This works well in constructed problems where you can easily see that a value like 0, 1, -1, 2 or -2 satisfies the equation.
3. Use the analog of the quadratic formula ($(-b\pm\sqrt{b^2-4ac})/2a$) for cubic equations. However it is messy, and unlike the quadratic formula hardy anyone knows it by heart.