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As is well known, the fundamental matrix $(I - P + e \pi)^{-1}$ plays animportant role in the performance analysis of Markov systems, where $P$ is thetransition probability matrix, $e$ is the column vector of ones, and $\pi$ isthe row vector of the steady state distribution. It is used to compute theperformance potential (relative value function) of Markov decision processesunder the average criterion, such as $g=(I - P + e \pi)^{-1} f$ where $g$ isthe column vector of performance potentials and $f$ is the column vector ofreward functions. However, we need to pre-compute $\pi$ before we can compute$(I - P + e \pi)^{-1}$. In this paper, we derive a generalization version ofthe fundamental matrix as $(I - P + e r)^{-1}$, where $r$ can be any given rowvector satisfying $r e \neq 0$. With this generalized fundamental matrix, wecan compute $g=(I - P + e r)^{-1} f$. The steady state distribution is computedas $\pi = r(I - P + e r)^{-1}$. The Q-factors at every state-action pair canalso be computed in a similar way. These formulas may give some insights onfurther understanding how to efficiently compute or estimate the values of $g$,$\pi$, and Q-factors in Markov systems, which are fundamental quantities forthe performance optimization of Markov systems.

A Generalized Fundamental Matrix for Computing Fundamental Quantities of Markov Systems