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. Furthermore, if is symmetric, then the columns of are orthogonal vectors.

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.