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In this letter, we consider an idea that induces flavor structure from inner products in noncommutative geometry. Assuming proper components of vectors $v_{(L,R) i}$ in enlarged representation space for fermions, we can induce the waterfall texture for Yukawa matrices retaining gauge interactions are universal. The hierarchy of the Yukawa interactions is a consequence of "misalignment" between the vectors $v_{Li}$ and $v_{Rj}$.

In this letter, we show a nilpotent matrix representation of the exterior derivative operator in noncommutative geometry (NCG), by translating the noncommutative relations of the algebraic formalization into the original one. As a result, we can construct the extended gauge theory without complicated algebraic rules nor excessively large differential algebra and non-monomorphic representation.