An eigenvalue is a special number associated with a square matrix in linear algebra. When a matrix acts on a vector, the eigenvalue indicates how much the vector is stretched or compressed. Specifically, if a vector is multiplied by a matrix and the result is a scaled version of the original vector, the scaling factor is the eigenvalue. Eigenvalues are important in various fields, including physics, engineering, and data science. They help in understanding systems' behaviors, such as stability in differential equations or dimensionality reduction in principal component analysis.