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In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter , in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is a generalization of Gillman's and Lezaud's analyses of the Ajtai-Komlos-Szemeredi sampler for real-valued functions [Gillman, 1993]. Derandomizing our sampler gives a few applications, yielding deterministic polynomial time algorithms for problems in which derandomizing independent sampling gives only quasi-polynomial time deterministic algorithms. The first (which was our original motivation) is to a polynomial-time derandomization of the Alon-Roichman theorem [Alon and Roichman, 1994]: given a group of size n, find O(log n) elements which generate it as an expander. This implies a second application - efficiently constructing a randomness-optimal homo-morphism tester, significantly improving the previous result of Shpilka and Wigderson . A third application, which derandomizes a generalization of the set cover problem, is deferred to the full version of this paper.