Quick way to find Eigenvalues

Formula

$${\lambda }_{1,2}=m\pm \sqrt{m^2-p}$$

Where $m\to$ Mean of the Eigen Values (diagonal) and $p\to$ product of the Eigen Values (determinant)

Example

Matrix

$$\left[\begin{array}{ll}3& 1\\ 4& 1\end{array}\right]$$

Mean $m=2$ Product $p=3-4=-1$ Eigenvalues

$$\boxed{\Rightarrow {\lambda }_1,{\lambda }_2=2\pm \sqrt{5}}$$

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Eigenvectors: Once we have the eigenvalues of a given matrix, find the eigenvectors by solving:

$$A\vec{v}=\lambda \vec{v}$$

which is equivalent to:

$$\begin{array}{c}(\mathbf{A}-\lambda \mathbf{I})\overrightarrow{\mathbf{v}}=\overrightarrow{0}\\ \text{ or }\\ (\lambda \mathbf{I}-\mathbf{A})\overrightarrow{\mathbf{v}}=\overrightarrow{0}\end{array}$$

The eigen vector can be found out by solving the matrix system. (Note: Here $\vec{v}=v_1{v}_2$)