100%(3)3 out of 3 people found this document helpful
This preview shows page 32 - 34 out of 68 pages.
Ux=cis solved by back-substitution. Here we concentrate on connectingAtoU.The matricesEfor step 1,Ffor step 2, andGfor step 3 were introduced in theprevious section. They are calledelementary matrices, and it is easy to see how theywork. To subtract a multipleof equationjfrom equationi,put the number−intothe(i,j)position. Otherwise keep the identity matrix, with 1s on the diagonal and 0selsewhere. Then matrix multiplication executes the row operation.
NOT FOR SALE Strang-5060bookApril 25, 200517:38331.5Triangular Factors and Row Exchanges33The result of all three steps isGFEA=U. Note thatEis the first to multiplyA,thenF, thenG. We could multiplyGFEtogether to find the single matrix that takesAtoU(and also takesbtoc). It is lower triangular (zeros are omitted):FromAtoUGFE=111111111−211=1−21−111.(3)This is good, but the most important question is exactly the opposite: How would we getfromUback toA?How can we undo the steps of Gaussian elimination?To undo step 1 is not hard. Instead of subtracting, weaddtwice the first row to thesecond. (Not twice the second row to the first!) The result of doing both the subtractionand the addition is to bring back the identity matrix:Inverse ofsubtractionis addition100210001100−210001=100010001.(4)One operation cancels the other. In matrix terms, one matrix is theinverseof the other.If the elementary matrixEhas the number−in the(i,j)position, then its inverseE−1has+in that position. ThusE−1E=I, which is equation (4).We can invert each step of elimination, by usingE−1andF−1andG−1. I think it’snot bad to see these inverses now, before the next section. The final problem is to undothe whole process at once, and see what matrix takesUback toA.Since step3was last in going from A to U, its matrix G must be the first tobe inverted in the reverse direction. Inverses come in the opposite order! The secondreverse step isF−1and the last isE−1:FromUback toAE−1F−1G−1U=AisLU=A.(5)You can substituteGFEAforU, to see how the inverses knock out the original steps.Now we recognize the matrixLthat takesUback toA. It is calledL, because it islower triangular. And it has a special property that can be seen only by multiplying thethree inverse matrices in the right order:E−1F−1G−1=121111−1111−11=121−1−11=L.(6)The special thing is thatthe entries below the diagonal are the multipliers=2,−1,and−1. When matrices are multiplied, there is usually no direct way to read off theanswer. Here the matrices come in just the right order so that their product can bewritten down immediately. If the computer stores each multiplieri j—the number thatmultiplies the pivot rowjwhen it is subtracted from rowi