Eigenvalue

An eigenvalue of a linear map $T : V \to V$ is a scalar $\lambda$ such that $T(\mathbf{v}) = \lambda \mathbf{v}$ for some nonzero vector $\mathbf{v}$, called an eigenvector corresponding to $\lambda$.

Eigenvalues are the roots of the characteristic polynomial $\det(A - \lambda I) = 0$. A linear map on a finite-dimensional space has at most $n$ eigenvalues (counted with multiplicity), where $n$ is the dimension.

Eigenvalues reveal the “natural scales” of a linear map: the directions along which the map acts by pure stretching or compression, without rotation.