Eigenvalues and eigenvectors

In linear algebra, an eigenvector ( EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector v {\displaystyle \mathbf {v} } of a linear transformation T {\displaystyle T} is scaled by a constant factor λ {\displaystyle \lambda } when the linear transformation is applied to it: ⁠ T v = λ v {\displaystyle T\mathbf {v} =\lambda \mathbf {v} } ⁠. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor λ {\displaystyle \lambda } (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, then the eigenvector's direction is reversed. The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all areas where linear algebra is applied, from geology to quantum mechanics. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation (feedback). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of the linear transformation, and the associated eigenvector is the steady state of the system.