factoriztionBasics

factoriztionBasics - FACTORING MATRICES TERRY A LORING A...

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FACTORING MATRICES TERRY A. LORING A lot of factorization results can be done using row operatations paired with column operations. To illustratate, suppose we have factored a 3 -by-3 matrix A as A = ST, for some 3 -by-3 matrices. We sort of like this factorization, but want a nicer one. We can insert I = 1 0 1 0 1 0 0 0 1 - 1 1 0 1 0 1 0 0 0 1 = 1 0 - 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 in the middle to get A = S 1 0 - 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 T . If we let S 1 = S 1 0 - 1 0 1 0 0 0 1 and T 1 = 1 0 1 0 1 0 0 0 1 T we have A = S 1 T 1 which might be better than the factorization we had before. This is a lot of writing of zeros. We know about how elementary matrices work when multiplied on the left of other matrices. They implement row operations. Multiplied on the right they implement column operations. If S = x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 then S 1 = x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 1 0 - 1 0 1 0 0 0 1 = x 1 y 1 z 1 - x 1 x 2 y 2 z 2 - x 2 x 3 y 3 z 3 - x 3 . 1

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2 TERRY A. LORING On the other hand, if T = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 then T 1 = 1 0 1 0 1 0 0 0 1 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 = a 1 + a 3 b 1 + b 3 c 1 + c 3 a 2 b 2 c 2 a 3 b 3 c 3 . That is, S is taken to S 1 via C 3 C 3 - C 1 while T is taken to T 1 via R 1 R 1 + R 3 .
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