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:

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

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 $\lambda$, 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:

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

We can find the determinant of the preceding matrix:

$det(A-\lambda I)=(-1)^{n}[\lambda ^{n}+c_{n-1}\lambda ^{n-1}...+c_{0}]$

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

Symmetric matrices

For any symmetric matrix, one can make an orthonormal basis out of their eigenvectors.