Let's say you are given the following easy matrix:
$$\begin{bmatrix}{-1}&{0}\\
{1}&{1}\end{bmatrix}$$
and you've calculated the following two eigenvalues:
$\lambda_{1} = -1$
$ \lambda_{2} = 1$
Is there a way to check whether the calculated eigenvalues are correct?
I know that with the eigenvectors you can just check everything all at once by checking it with this formula:
$$A. \vec{x} = \lambda \vec{x}$$
But how can you check the correctness of the computed results without having to calculate all the eigenvectors and fill in the formula?
Best Answer
Suppose you have a $2\times 2$ matrix, A. You can use the fact that $$\lambda^2 -\lambda\tau(A)+\det(A)=0$$ Where $\tau(A)$ is the trace of your matrix $A$.