Let P be a matrix of eigenvectors of a given square matrix A and D be a diagonal matrix with the corresponding eigenvalues on the diagonal. Then, as long as P is a square matrix, A can be written as an eigen decomposition A = P D P^(-1), where D is a diagonal matrix. Furthermore, if A is symmetric, then the columns of P are orthogonal vectors. If P is not a square matrix (for example, the space of eigenvectors of [1 | 1 0 | 1] is one-dimensional), then P cannot have a matrix inverse and A does not have an eigen decomposition. However, if P is m×n (with m>n), then A can be written using a so-called singular value decomposition.
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