Solution of Rectangular Systems of Linear Equations Using Orthogonalization and Projection Matrices

Abstract

IN THIS PAPER A NOVEL APPROACH TO THE SOLUTION OF RECTANGULAR SYSTEMS OF LINEAR EQUATIONS IS PRESENTED. IT STARTS WITH A HOMOGENEOUS SET OF EQUATIONS AND THROUGH LINEAR SE SPACE CONSIDERATIONS OBTAINS THE SOLUTION BY FINDING THE NULL SPACE OF THE COEFFICIENT MATRIX. TO DO THIS AN ORTHOGONAL BASIS FOR THE ROW SPACE OF THE COEFFICIENT MATRIX IS FOUND AND THIS BASIS IS COMPLETED FOR THE WHOLE SPACE USING THE GRAM""SCHMIDT ORTHOGONALIZATION PROCESS. THE NON HOMOGENEOUS CASE IS HANDLED BY CONVERTING THE PROBLEM INTO A HOMOGENEOUS ONE, PASSING THE RIGHT SIDE VECTOR TO THE LEFT SIDE, LET TING THE COMPONENTS OF THE NEGATIVE OF THE RIGHT SIDE BE COME THE COEFFICIENTS OF AND ADDITIONAL VARIABLE, SOLVING THE NEW SYSTEM AND AT THE ENDIMPOSING THE CONDITION THAT THE ADDITIONAL VARIABLE TAKE A UNIT VALUE. IT IS SHOWN THAT THE NULL SPACE OF THE COEFFICIENT MATRIX IS INTIMATELY CONNECTED WITH ORTHOGONAL PROJECTION MATRICES WHICH ARE EASILY CONSTRUCTED FROM THE ORTHOGONAL BASIS USING DYADS. THE PAPER TREATS THE METHOD INTRODUCED AS AN EXACT METHOD WHEN THE ORIGINAL COEFFICIENTS ARE RATIONAL AND RATIONAL ARITHMETIC IS USED. THE ANALYSIS OF THE EFFICIENCY AND NUMERICAL CHARACTER IS TICS OF THE METHOD IS DEFERRED TO A FUTURE PAPER. DE TAILED NUMERICAL ILLUSTRATIVE EXAMPLES ARE PROVIDED IN THE PAPER AND THE USE OF THE PROGRAM MATHEMATICA TO PERFORM THE COMPUTATIONS IN RATIONAL ARITHMETIC IS ILLUSTRATED