We introduce a new class of simultaneously diagonalizable real matrices, the γ -matrices, which include both symmetric circulant matrices and a subclass of the set of all reverse circulant matrices. We define some algorithms for fast computation of the product between a γ -matrix and a real vector. We proved that the computational cost of a multiplication between a γ -matrix and a real vector is of at most 74nlog2n+o(nlog2n) additions and 12nlog2n+o(nlog2n) multiplications. Our algorithm can be used to improve the performance of general discrete transforms for multiplications of real vectors.
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