Jordan decomposition is a mathematical concept used in linear algebra to analyze linear transformations. It states that any square matrix can be expressed in a specific form, known as the Jordan normal form, which simplifies the study of its properties. This decomposition reveals the structure of the matrix by breaking it down into simpler components, making it easier to understand its behavior. In Jordan normal form, a matrix is represented as a block diagonal matrix, where each block corresponds to an eigenvalue of the original matrix. These blocks can be either Jordan blocks or zero matrices, depending on the algebraic and geometric multiplicities of the eigenvalues. This decomposition is particularly useful for solving systems of differential equations and studying the stability of dynamical systems.