Eigenvector/Eigenvalue

Eigenvectors are aka. eigenfunctions

https://www.youtube.com/watch?v=PFDu9oVAE-g

\begin{gather*} A\vec{v}=\lambda\vec{v} \end{gather*}

Basically if I apply matrix transformation $A$ to a vector $\left\lvert v \right\rangle$ and it still points in the same direction, it is an Eigenvector. It could be multiplied by a scalar value $\lambda$ (Eigenvalue) after applying the transform, which means its Magnitude need not be preserved by the transform.

The Set of all vector $\left\lvert v \right\rangle$ ’s are Eigenvectors of transformation $A$ .

An Eigenstate are states instead of Vector, Operator instead of Matrix but are the same thing.