Eigen Decomposition Theorem

Let be a nonsingular matrix whose columns are eigenvectors of a given square matrix , and let be a diagonal matrix with the corresponding eigenvalues on the diagonal. Then can be written as an eigen decomposition

where is a diagonal matrix. If is a real symmetric matrix, can be chosen to be an orthogonal matrix. More generally, for a Hermitian matrix, it can be chosen to be a unitary matrix. In these cases, eigenvectors belonging to distinct eigenvalues are orthogonal vectors, while within a repeated eigenvalue an orthonormal basis of the eigenspace is chosen. An arbitrary vector basis of that eigenspace need not be orthogonal (Axler 2024, pp. 245-246).

If does not have enough linearly independent eigenvectors to form such a matrix (for example, the space of eigenvectors of is one-dimensional), then cannot have a matrix inverse and does not have an eigen decomposition. However, every real matrix can be written using a so-called singular value decomposition.