Eigenvector

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

where is the eigenvalue associated with the eigenvector

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

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 , the eigenspace is all the eigenvectors with that particular eigenvalue.

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:

We can find the determinant of the preceding matrix:

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