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 }

Eigenvectors are the foundation of the diagonalization technique.

Intuition

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

Eigenspace

Given A and 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} , the eigenspace is all the eigenvectors.

Because any linear combination of eigenvectors will yield another eigenvector of the same eigenvalue, the eigenspace is a vector space.

The following definition of eigenvectors help explore the eigenspace:

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 }

We can find the determinant of the preceding matrix:

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 det(A-\lambda I)=(-1)^n[\lambda^n+c_{n-1}\lambda^{n-1}...+c_0] }

This is the characteristic polynomial of the A, showing that for any A, there are at most n eigenvalues.