Eigenvalues, represented by the Greek letter lambda (λ), are a fundamental concept in linear algebra that are associated with square matrices (arrays of numbers arranged in rows and columns), which arise from the linear transformation of a vector space (a collection of vectors), and can be found by solving the characteristic equation (the determinant of the difference between the original matrix and a multiple of the identity matrix, which is a square matrix with ones on the diagonal and zeros elsewhere), where the determinant is a scalar value derived from the elements of a matrix representing the signed volume scaling factor of the linear transformation, and once we have the eigenvalues, we can determine the eigenvectors (non-zero vectors that only change by a scalar multiple when a linear transformation is applied to them) by plugging each eigenvalue back into the modified characteristic equation and solving for the vector, with the eigenvalue-eigenvector pairs providing important information about the transformation, such as its stretch or compression factors, rotational effects, and whether it is diagonalizable (able to be transformed into a diagonal matrix by a similarity transformation), which in turn has applications in various fields including physics, engineering, computer science, and economics, where eigenvalues and eigenvectors are used to analyze stability, solve differential equations, find the principal components in data analysis, and perform spectral clustering, among many other applications.