What are eigenvalues and eigenvectors?

by Amrita Mitra | Oct 3, 2023 | Featured, Linear Algebra

$\begin{bmatrix} -2 & 2 \\ 2 & -2 \end{bmatrix} \cdot \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}=0 \\ or,\ -2u_1+2u_2=0\ and\ 2u_1-2u_2=0$

By solving these two equations, we get

$u_1=1\ and\ u_2=1$

Similarly, we can write:

$\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix} \cdot \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}=0 \\ or,\ 2v_1+2v_2=0$

So, one solution of the given equation is:

$v_1=-1\ and\ v_2=1$

Therefore, we get the following two eigenvectors for the given eigenvalues:

$\vec{u}=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix}\ and\ \\ \vec{v}=\frac{1}{\sqrt{2}} \begin{bmatrix} -1 \\ 1 \end{bmatrix}$

Please note that the eigenvectors have a magnitude of 1.

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