Eigenvector

Given a matrix, its eigenvectors are special vectors that satisfy the following property:

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

where $\lambda$ is the eigenvalue associated with the eigenvector ${\vec {x}}$

The definition of eigenvectors are also frequently written in this form:

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

Intuition

If we think of a matrix as a linear transformation, eigenvectors do not change direction. Instead, they simply scale by an eigenvalue.