# What's the Nature of Eigenvector?

By Xah Lee. Date:

What's Eigenvalues and eigenvectors?

An eigenvector of a square matrix A is a non-zero vector v that, when the matrix is multiplied by v, yields a constant multiple of v, the multiplier being commonly denoted by λ. That is:

`A v = λ v`

The number λ is called the eigenvalue of A corresponding to v.

here's a funtional style of description.

eigenvector and eigenvalue are special elements associated with some linear function of vector space.

Let f:𝕍→𝕍 be a linear function on vector space 𝕍. A eigenvector of f is any none-zero element v in 𝕍 such that

`f[v] == λ*v`

for some constant λ. λ is called the eigenvalue for the eigenvector v.

here's a illustration of eigenvector.

eigenvector is also called characteristic vector because it characterize the linear function.

not all linear function has eigenvectors. For example, rotation around origin (on 2D vectors) is a linear function, but doesn't have eigenvector. See also: Nature of Linear Transformation.