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

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

2011-12-07

arxiv.org/abs/1112.1588v2

Mathematics

Quantum Algebra

23 pages, light version submitted to SIAM Journal on Matrix Analysis and Applications (SIMAX). A separate paper with the graph

Scientific paper

We introduce the method of path-sums which is a tool for exactly evaluating a function of a discrete matrix with possibly non-commuting entries, based on the closed-form resummation of infinite families of terms in the corresponding Taylor series. If the matrix is finite, 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, matrix inverse, and matrix logarithm. We show that the quasideterminants of a matrix can be naturally formulated in terms of a path-sum, and present examples of the application of the path-sum method. We show that obtaining the inversion height of a matrix inverse and of quasideterminants is an NP-complete problem.

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