Abstract
We introduce the method of path-sums, which is a tool for analytically evaluating a primary function of a finite square discrete matrix based on the closed-form resummation of infinite families of terms in the corresponding Taylor series. Provided the required inverse transforms are available, our approach yields the exact result in a finite number of steps. We achieve this by combining a mapping between matrix powers and walks on a weighted directed graph with a universal graph-theoretic result on the structure of such walks. We present path-sum expressions for a matrix raised to a complex power, the matrix exponential, the matrix inverse, and the matrix logarithm. We present examples of the application of the path-sum method.
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