have an additional freedom of linear transformation, that is to say, any linear (orthonormal) combination of eigenvectors sharing an eigenvalue (in the degenerate subspace) is itself an eigenvector (in the subspace). Useful facts regarding eigendecomposition can be eigendecomposed if and only if the number of linearly independent eigenvectors, , equals the dimension of an eigenvector: If the field of scalars is algebraically closed and if has no repeated roots, that is, if then can be eigendecomposed. The statement " can be eigendecomposed" does not imply that has an inverse as some eigenvalues may be zero, which is not invertible. The statement " has an inverse" does not imply that can be eigendecomposed. A counterexample is , which is an invertible defective matrix. Useful facts regarding matrix inverse can be inverted if and only if all eigenvalues are nonzero: If and , the inverse is given by Numerical computations Numerical computation of eigenvalues Suppose that we want to compute the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the characteristic polynomial. However, this is often impossible for larger matrices, in which case we must use a numerical method. In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Computing the polynomial becomes expensive in itself, and exact (symbolic) roots of a high-degree polynomial can be difficult to compute and express: the Abel–Ruffini theorem implies that the roots of high-degree (5 or above) polynomials cannot in general be expressed simply using th roots. Therefore, general algorithms to find eigenvectors and eigenvalues are iterative. Iterative numerical algorithms for approximating roots of polynomials exist, such as Newton's method, but in general it is impractical to compute the characteristic polynomial and then apply these methods. One reason is that small round-off errors in the coefficients of the characteristic polynomial can lead to large errors in the eigenvalues and eigenvectors: the roots are an extremely ill-conditioned function of the coefficients. A simple and accurate iterative method is the power method: a random vector is chosen and a sequence of unit vectors is computed as This sequence will almost always converge to an eigenvector corresponding to the eigenvalue of greatest magnitude, provided that has a nonzero component of this eigenvector in the eigenvector basis (and also provided that there is only one eigenvalue of greatest magnitude). This simple algorithm is useful in some practical applications; for