We study the computational complexity of a diagonalization technique for multivariate homogeneous polynomials, expressing them as the sums of powers of independent linear forms. It is based on Harrison's center theory and consists of a criterion and a diagonalization algorithm. Detailed formulations and the computational complexity of each component of the technique are given. The complexity analysis focuses on the impacts of the number of variables and the degree of given polynomials. We show that this criterion runs in polynomial time, and the diagonalization process performs efficiently in numerical experiments. Other diagonalization techniques are reviewed and compared in terms of complexity.