Abstract
The Moore-Penrose inverse is the most popular type of matrix generalized inverses which has many applications both in matrix theory and numerical linear algebra. It is well known that the Moore-Penrose inverse can be found via singular value decomposition. In this regard, there is the most effective algorithm which consists of two stages. In the first stage, through the use of the Householder reflections, an initial matrix is reduced to the upper bidiagonal form (the Golub-Kahan bidiagonalization algorithm). The second stage is known in scientific literature as the Golub-Reinsch algorithm. This is an iterative procedure which with the help of the Givens rotations generates a sequence of bidiagonal matrices converging to a diagonal form. This allows to obtain an iterative approximation to the singular value decomposition of the bidiagonal matrix. The principal intention of the present paper is to develop a method which can be considered as an alternative to the Golub-Reinsch iterative algorithm. Realizing the approach proposed in the study, the following two main results have been achieved. First, we obtain explicit expressions for the entries of the Moore-Penrose inverse of bidigonal matrices. Secondly, based on the closed form formulas, we get a finite recursive numerical algorithm of optimal computational complexity. Thus, we can compute the Moore-Penrose inverse of bidiagonal matrices without using the singular value decomposition.
Highlights
As is known, for a real m × n matrix A the Moore-Penrose inverse A+ is the unique matrix that satisfies the following four properties [1]: AA+A = A, A+AA+ = A+,(A+A)T = A+A, (AA+)T = AA+ .If A is a square nonsingular matrix, A+ = = A−1
There is well-known formula for the MoorePenrose inverse which is obtained by the singular value decomposition of the matrix
The singular value decomposition of an m × n matrix A with rank r is its factorization of the form
Summary
Thereby the problem is reduces to the Moore-Penrose inversion of the bidiagonal matrix (1.3). Once the bidiagonalization of the initial matrix has been achieved, the task is to zero the superdiagonal entries in the matrix (1.3) With this purpose the Golub-Reinsch algorithm is implemented [3]. Having the SVD, the Moore-Penrose inverse of the matrix is computed (see [1; 4], for instance). The objective of the present work is to develop a method which allows to deduce formulas for the entries of the Moore-Penrose inverse of upper bidiagonal matrices. The obtained closed form solution to the Moore-Penrose inversion may be considered as an alternative to sufficiently labour-consuming Golub-Reinsch iterative procedure briefly described in the Stage 2 of this section.
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