In this paper we propose a new representation for FFT algorithms called the triangular matrix representation. This representation is more general than the binary tree representation and, therefore, it introduces new FFT algorithms that were not discovered before. Furthermore, the new representation has the advantage that it is simple and easy to understand, as each FFT algorithm only consists of a triangular matrix. Besides, the new representation allows for obtaining the exact twiddle factor values in the FFT flow graph easily. This facilitates the design of FFT hardware architectures. As a result, the triangular matrix representation is an excellent alternative to represent FFT algorithms and it opens new possibilities in the exploration and understanding of the FFT.