# What's the Nature of Eigenvector?

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 vThe 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.

- On the Naming of Eigenvector and the Igon Value Problem
- the Nature of Associative Property of Algebra
- The Geometric Significance of Complex Conjugate
- The Significance of Complex Numbers: Frobenius Theorem
- What is Riemann Surface? Understanding the Concept Without Math
- Geometry: Transformation of the Plane II