Let be a matrix of eigenvectors of a given square matrix and be a diagonal matrix with the corresponding eigenvalues on the diagonal. Then, as long as is a square matrix, can be written as an eigen decomposition
where is a diagonal matrix. Furthermore, if is symmetric, then the columns of are orthogonal vectors.
If is not a square 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, if is (with ), then can be written using a so-called singular value decomposition.