Applied Linear Statistical ModelsGaussian Elimination for Finding the InverseDr. DR JonesSweep OperatorWhere A is an r x r square matrix and k is the index of the row (and column) to be sweptSWP (A, k) = BB is the resulting r x r matrixHow the Sweep Operator WorksThe operation may not be performed if the fipivot,fl akk, equals zero.ExampleThe sweep operator can compute A-1:ExampleThe Sweep Operation on Partitioned MatricesA Partitioning of AWhere A11 is a p x p matrix,SWP Operation is Performed on the p Pivot Elements in A11Resulting MatrixSweep Operator Computes Solutions to Systems of Linear EquationsSweep Operations Computes the Determinant of a Matrix:Test for Singular MatrixIf Sweep encounters zero pivotsMatrix is singular (does not have an inverse)The determinant is zeroRank and Positive Definite MatrixThe rank of a matrix is the number of non-zero pivotsIf all pivots are positive then the determinant is positive and the matrix is said to be positive definite.ExampleExample ContinuedExample ContinuedRegression ExampleThe Least Squares Formulas Using Sweep OperationsSweep operations using Toluca dataApplied Linear Statistical ModelsGaussian Elimination for Finding the InverseDr. DR Jones Sweep OperatorWhere A is an r x r square matrix and k is the index of the row (and column)to be sweptSWP (A, k) = BB is the resulting r x r matrixHow the Sweep Operator Worksbaaaaij kbaajkbaaikbaij ijik klkkkjkjkkikikkkkkkk=− ≠=≠=− ≠=for all ,.10The operation may not be performed if the “pivot,” akk, equals zero.Example()ABA===−−⋅=−−2431112423213425215 5SWP ,..The sweep operator can compute A-1:()() ( )()()AAAAAA01112====−−kkrSWP k k r,,,...,Example( )( )( )( ) ( )( )( )01211243114.5 222,1334 1.551221.5 2 2.5 3.0 22.55555,21.5 11.5 15555.1 .4.3 .2SWPSWP− =     = = =  ⋅ − −  − −   − ⋅ − + − −  −− − = = = −  −   − − − = = − AAAAAAThe Sweep Operation on Partitioned MatricesA Partitioning of AAAAAA=11 1221 22Where A11 is a p x p matrix,SWP Operation is Performed on the p Pivot Elements in A11() ( )()AAkkSWP k k p==−112,,,...,Resulting Matrix()AAAAAA A AAAp=−−−−−−1111111221 11122 21 11112Sweep Operator Computes Solutions to Systems of Linear Equations( )11111 12111122 21 11 1221 12Ifp−−−== = ′− −  ′=12AB ABAAABABA AAAAASweep Operations Computes the Determinant of a Matrix:( )( ) ( ){ }( )1 the pivot is diagonal element of kthkkkkthkkrkkkrrkakakDET a×==∏AATest for Singular MatrixIf Sweep encounters zero pivotsMatrix is singular (does not have an inverse)The determinant is zeroRank and Positive Definite MatrixThe rank of a matrix is the number of non-zero pivotsIf all pivots are positive then the determinant is positive and the matrix issaid to be positive definite.Example01000111 12011 12124536Solution: 1.9; 0.324 531 6bbbbbbbb+=+==== = = =  AB AAABExample Continued( )( ) ( )( )( ) ( )( )01021111 1224 531 65670.5 2 2.5,1 1.5 5 1.52.5 4 57.5.1 .4 1.9,2 .3 .2 .31.3 .8 58.71.9.3SWPSWP−  =     = = − − −  − −  − = = −  − −  = =  AAAAABAAExample Continued{}()rank non zeroDETAA1122112225 225 10××=− −==⋅− =−#,Regression Example()′=′==′=′=′′∑∑∑∑∑−XX XYbXXbXYbXXXYnXXXYXYbbiiiiii2011The Least Squares Formulas Using Sweep Operations( )( )( )( ) ( )( )0221120211,111,2iiii iiiiiiXX XYXY YYXYXX XX XXXYXX XX XXXY XYYYXX XXnXYXX XYYXYYnXYSWP X SS SSYSS SSXX SSXbYnSS SS SSXSSSWP bSS SS SSSS X SS SSY SSE SSSS SS− ′ ′  = =  ′ ′     = = −  − −+ = −−= = =− = −∑ ∑∑ ∑ ∑∑ ∑ ∑XX XYAYX YYAAAA2XYXXSS           Sweep operations using Toluca data( )( )( )( ) ( )( )012125 1,750 7,8071,750 142,300 617,1807,807 617,180 2,745,173.04 70 312.28,1 70 19,800 70,690312.28 70,690 307,203.287475 .00345 62.36586,2 .00345 5.05 05 3.57020262.36586 3.5SWPSWP E  =     = = −  − −= = − −− −AAAAA70202 54,825.46    