Eigenvector

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

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is the eigenvalue associated with the eigenvector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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.