vol18_pp352-363

vol18_pp352-363 - Electronic Journal of Linear Algebra ISSN...

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ELA FULL RANK FACTORIZATION AND THE FLANDERS THEOREM RAFAEL CANT ´ O , BEATRIZ RICARTE , AND ANA M. URBANO Abstract. In this paper, a method is given that obtains a full rank factorization of a rectangular matrix. It is studied when a matrix has a full rank factorization in echelon form. If this factorization exists, it is proven to be unique. Applying the full rank factorization in echelon form the Flanders theorem and its converse in a particular case are proven. Key words. Echelon form of a matrix, LU factorization, Full rank factorization, Flanders theorem. AMS subject classifcations. 15A15, 15A23. 1. Introduction. Triangular factorizations of matrices play an important role in solving linear systems. It is known that the LDU factorization is unique for square nonsingular matrices and for full row rank rectangular matrices. In any other case, the LDU factorization is not unique and the orders of L , D and U are greater than the rank of the initial matrix. We focus our attention on matrices A R n × m with rank( A )= r min { n,m } , where the LDU factorization of A is not unique. For this kind of matrices, it is useful to consider the full rank factorization of A , that is, a decomposition in the form A = FG with F R n × r , G R r × m and rank( F )=rank( G r . The full rank factorization of any nonzero matrix is not unique. In addition, if A = is a full rank factorization of A , then any other full rank factorization can be written in the form A =( FM 1 )( MG ), where M R r × r is a nonsingular matrix. If the full rank factorization of A is given by A = LDU ,where L R n × r is in lower echelon form, D = diag( d 1 ,d 2 ,...,d r ) is nonsingular and U R r × m is in upper echelon form, then this factorization is called a full rank factorization in echelon form of A . In this paper we give a method to obtain a full rank factorization of a rectangular matrix and we study when this decomposition can be in echelon form. Moreover, if the factorization in echelon form exists, we prove that it is unique. Finally, applying Received by the editors January 21, 2009. Accepted for publication July 9, 2009. Handling Editor: Joao Filipe Queiro. Institut de Matem` atica Multidisciplinar, Universidad Polit´ ecnica de Valencia, 46071 Valencia, Spain ([email protected], [email protected], [email protected]). Supported by the Spanish DGI grant MTM2007-64477 and by the UPV under its research program. 352 Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 352-363, July 2009 http://math.technion.ac.il/iic/ela
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ELA Full Rank Factorization and the Flanders Theorem 353 the full rank factorization in echelon form, we give a simple proof of the Flanders theorem [4] for matrices A R n × r and B R r × n with rank( A )=rank( B )= r ,as well as of the converse result.
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vol18_pp352-363 - Electronic Journal of Linear Algebra ISSN...

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