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An efficient algorithm for the computation of powers of a square arbitrary lower triangular matrix is presented. A comparison of the algorithm with the standard matrix multiplication method in terms of maximum number of multiplications is included. A numerical example is used to show some of the computational details.

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The main object of this note is to show that a matrix of order 3 can be triangularized by just one sequence of rotations. In [1] we describe a process to triangularize a matrix of order 3. The process consists of two steps. The first step is to apply a sequence of two Greenstadt's rotations [1] with pivot-pairs (1, 2) and (2, 3). This sequence of rotations shall either triangularize the matrix, or the resulting matrix becomes the exceptional form (see equation (8) of [1]). The second step of the process wants to change the matrix from the exceptional form to the non-exceptional so that the first step can resume its function again. Parlett and Wang [2] point out that it probably would take the first step an infinite sequence of rotations to bring the matrix to the exceptional form. Therefore, in order to triangularize the matrix, the process would have to apply a concatenation of infinitely many infinite sequences of rotations. Wang [2] proposes a way to eliminate this concatenation by trying to apply the second step prior to the "end" of the first step, i.e., prior to some point where the matrix is turning into the exceptional form. In this note we show that the process described in [1] does not require a concatenation of many infinite sequences of rotations, it only requires one sequence of rotations to triangularize a matrix. Here we consider only the matrix of order 3, but we believe that a similar analysis can also be applied to matrices of higher order. Throughout this note we assume that notations that are used without specification are the same notations used in [1].

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