diagonal matrix from the eigendecomposition of is thus . Putting the solutions back into the above simultaneous equations Solving the equations, we have Thus the matrix required for the eigendecomposition of is that is: Matrix inverse via eigendecomposition If a matrix can be eigendecomposed and if none of its eigenvalues are zero, then is invertible and its inverse is given by If is a symmetric matrix, since is formed from the eigenvectors of , is guaranteed to be an orthogonal matrix, therefore . Furthermore, because is a diagonal matrix, its inverse is easy to calculate: Practical implications When eigendecomposition is used on a matrix of measured, real data, the inverse may be less valid when all eigenvalues are used unmodified in the form above. This is because as eigenvalues become relatively small, their contribution to the inversion is large. Those near zero or at the "noise" of the measurement system will have undue influence and could hamper solutions (detection) using the inverse. Two mitigations have been proposed: truncating small or zero eigenvalues, and extending the lowest reliable eigenvalue to those below it. See also Tikhonov regularization as a statistically motivated but biased method for rolling off eigenvalues as they become dominated by noise. The first mitigation method is similar to a sparse sample of the original matrix, removing components that are not considered valuable. However, if the solution or detection process is near the noise level, truncating may remove components that influence the desired solution. The second mitigation extends the eigenvalue so that lower values have much less influence over inversion, but do still contribute, such that solutions near the noise will still be found. The reliable eigenvalue can be found by assuming that eigenvalues of extremely similar and low value are a good representation of measurement noise (which is assumed low for most systems). If the eigenvalues are rank-sorted by value, then the reliable eigenvalue can be found by minimization of the Laplacian of the sorted eigenvalues: where the eigenvalues are subscripted with an to denote being sorted. The position of the minimization is the lowest reliable eigenvalue. In measurement systems, the square root of this reliable eigenvalue is the average noise over the components of the system. Functional calculus The eigendecomposition allows for much easier computation of power series of matrices. If is given by then we know that Because is a diagonal matrix, functions of are very