# What is the Nature of Eigenvector?

What is [ Eigenvalues and eigenvectors ] [ https://en.wikipedia.org/wiki/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 .

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