have index equal to ; that is, its minimal polynomial has only simple roots. Note that knowing a matrix's spectrum with all of its algebraic/geometric multiplicities and indexes does not always allow for the computation of its Jordan normal form (this may be a sufficient condition only for spectrally simple, usually low-dimensional matrices): the Jordan decomposition is, in general, a computationally challenging task. From the vector space point of view, the Jordan decomposition is equivalent to finding an orthogonal decomposition (that is, via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for. Functions of matrices Let (that is, a complex matrix) and be the change of basis matrix to the Jordan normal form of ; that is, . Now let be a holomorphic function on an open set such that ; that is, the spectrum of the matrix is contained inside the domain of holomorphy of . Let be the power series expansion of around , which will be hereinafter supposed to be 0 for simplicity's sake. The matrix is then defined via the following formal power series and is absolutely convergent with respect to the Euclidean norm of . To put it another way, converges absolutely for every square matrix whose spectral radius is less than the radius of convergence of around and is uniformly convergent on any compact subsets of satisfying this property in the matrix Lie group topology. The Jordan normal form allows the computation of functions of matrices without explicitly computing an infinite series, which is one of the main achievements of Jordan matrices. Using the facts that the th power () of a diagonal block matrix is the diagonal block matrix whose blocks are the th powers of the respective blocks; that is, and that , the above matrix power series becomes where the last series need not be computed explicitly via power series of every Jordan block. In fact, if , any holomorphic function of a Jordan block has a finite power series around because . Here, is the nilpotent part of and has all 0's except 1's along the superdiagonal. Thus it is the following upper triangular matrix: As a consequence of this, the computation of any function of a matrix is straightforward whenever its Jordan normal form and its change-of-basis matrix are known. For example, using , the inverse of is: