One important issue for matrix eigenvalue problems on high performance parallel computers is the cost of data movement. The paper shows that the eigenvalues and eigenvectors of a symmetric \(n\times n\) matrix can be found by \(O(\log_ 2n)\) matrix multiplications. A numerical method is developed to reduce the symmetric matrix eigenvalue problem to a number of matrix multiplications. The principle of the method is based on the fact that matrix multiplications can be implemented more efficiently than matrix-vector or vector-vector operations. The method performs \(O(n^ 3\log_ 2n)\) floating point operations with \(O(n^ 2\log_ 2n)\) data movement, while the traditional algorithm performs \(O(n^ 3)\) operations and \(O(n^ 3)\) data movements.