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Dimension Reduction (DR) algorithms are generally categorized into feature extraction and feature selection algorithms. In the past, few works have been done to contrast and unify the two algorithm categories. In this work, we introduce a matrix trace oriented optimization framework to provide a unifying view for both feature extraction and selection algorithms. We show that the unified view of DR algorithms allows us to discover some essential relationships among many state-of- the-art DR algorithms. Inspired by these essential insights, we propose to synthesize unlimited number of novel DR algorithms by combining, mapping and integrating the state- of-the-art algorithms. We present examples of newly synthesized DR algorithms with experimental results to show the effectiveness of our automatically synthesized algorithms.