How to check the correctness of calculated eigenvalues? 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?
 A: 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 $A$.
A: For a $2\times 2$ matrix $A$, the characteristic polynomial equation defining the eigenvalues is given by [https://en.wikipedia.org/wiki/Characteristic_polynomial]
$$
\lambda^2-(\mathrm{Tr}A)\lambda+\det(A)=0\ ,
$$
where $A$ is the matrix trace (the sum of diagonal entries $a_{11}+a_{22}$), and $\det(A)$ is its determinant $a_{11}a_{22}-a_{12}a_{21}$. If you solve this equation, its two roots $\lambda_{1,2}$ are the eigenvalues of $A$. It follows from the general theory of quadratic equation that $\lambda_1+\lambda_2$ is equal to the coefficient of the $\lambda$ term (with a minus sign), while the product $\lambda_1\lambda_2$ is equal to the zeroth order (constant) term of the equation. Therefore, if you compute somehow two eigenvalues (call them $\mu_1,\mu_2$) and want to check whether they are correct or not, you need to verify that $\mu_1+\mu_2=a_{11}+a_{22}$ and $\mu_1\mu_2=a_{11}a_{22}-a_{12}a_{21}$. These two equations are sufficient to tell you with certainty whether your postulated eigenvalues are OK or not - but you need both conditions - you might easily engineer situations where the 'trace' test works, but the 'det' test fails (which means your postulated eigenvalues are wrong). 
