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 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.
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