On the triangularization of arbitrary matrices

Abstract

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].

The implementation of a Jacobi-type method

“…In [2,3] we describe and prove the convergence of a Jacobi-type procedure for the triangularization of an arbitrary matrix. The triangularization procedure requires that in order to triangularize a matrix of order n, its two principal submatrices of order (n-l) have to be triangularized first.…”

“…The rotation V(l,n) will make some of the below diagonal elements of the two principal submatrices nonzero, and hence they have to be triangularized again. Theoretically, it takes an infinite series of plane rotations to triangularize a matrix of order 3 [3], and hence it needs an infinite sequence of infinite series of plane rotations to triangularize a matrix of order n for n > 3. Also switching back and forth among submatrices of lower order is timeconsuming if the procedure is implemented on computers.…”

“…Re(X) is the real part of the complex number X.It is shown in[2,3] that as long as the quantity inside the square bracket is non-zero, then there exist @ and m such that ~01 is greater than zero. Now if both the quantity inside the square bracket is zero, and the quantity underlined in equation (2.3) is positive or equal to zero, then there exist no such @ and…”