What are eigenvalues and eigenvectors?

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

$A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$

We can get the eigen values of A in the following way:

$det(A- \lambda I)=0 \\ or,\ \begin{bmatrix} 1- \lambda & 2 \\ 2 & 1- \lambda \end{bmatrix}=0 \\ or,\ (1- \lambda)^2-4=0 \\ or,\ \lambda=3,-1$

So, we can say that,

$(A- \lambda I) \vec{u}=0$

where λ is an eigenvalue of A and u is an eigenvector of A.

If we put the values of A and λ, we get:

$\begin{bmatrix} 1- \lambda & 2 \\ 2 & 1- \lambda \end{bmatrix} \cdot \vec{u}=0$

So,

$\begin{bmatrix} 1-3 & 2 \\ 2 & 1-3 \end{bmatrix} \cdot \vec{u}=0 \\ or,\ \begin{bmatrix} -2 & 2 \\ 2 & -2 \end{bmatrix} \cdot \vec{u}=0\ when\ \lambda =3$

And,

$\begin{bmatrix} 1-(-1) & 2 \\ 2 & 1-(-1) \end{bmatrix} \cdot \vec{u}=0 \\ or,\ \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix} \cdot \vec{u}=0\ when\ \lambda =-1$

So, we can now solve for the above two equations to get the eigenvectors.

$\begin{bmatrix} -2 & 2 \\ 2 & -2 \end{bmatrix} \cdot \vec{u}=0 \\ \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix} \cdot \vec{v}=0$

Now, let’s say u and v are the following two vectors:

$\vec{u}=\begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \\ \vec{v}=\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$

So, we can say that, …

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