An eigenvector is a special type of vector in linear algebra that, when a linear transformation is applied to it, does not change its direction. Instead, it may only be scaled by a factor known as the eigenvalue. This concept is crucial in various fields, including physics, computer science, and statistics, as it helps simplify complex problems. In mathematical terms, if a matrix A acts on an eigenvector v, the result is the same as multiplying v by a scalar λ (the eigenvalue). This relationship can be expressed as Av = λv. Eigenvectors and eigenvalues are essential in understanding systems of equations, stability analysis, and data reduction techniques like Principal Component Analysis (PCA).