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We introduce the method of path-sums which is a tool for exactly evaluating afunction of a discrete matrix with possibly non-commuting entries, based on theclosed-form resummation of infinite families of terms in the correspondingTaylor series. If the matrix is finite, our approach yields the exact result ina finite number of steps. We achieve this by combining a mapping between matrixpowers and walks on a weighted directed graph with a universal graph-theoreticresult on the structure of such walks. We present path-sum expressions for amatrix raised to a complex power, the matrix exponential, matrix inverse, andmatrix logarithm. We show that the quasideterminants of a matrix can benaturally formulated in terms of a path-sum, and present examples of theapplication of the path-sum method. We show that obtaining the inversion heightof a matrix inverse and of quasideterminants is an NP-complete problem.

Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums