Eigenvalues are special numbers associated with a square matrix in linear algebra. They represent the factors by which a corresponding eigenvector is stretched or compressed during a linear transformation. When a matrix acts on an eigenvector, the output is simply the eigenvector scaled by its eigenvalue. To find the eigenvalues of a matrix, one typically solves the characteristic equation, which is derived from the determinant of the matrix minus a scalar multiple of the identity matrix. The solutions to this equation provide the eigenvalues, which are crucial in various applications, including physics, engineering, and data analysis.