Lecture6.pdf - LECTURE 6 §  LU factorization §  LDU factorization §  Uniqueness §  LU factorization of non-square matrices A little detour

Lecture6.pdf - LECTURE 6 §  LU factorization §  LDU...

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LECTURE 6 § LU factorization § LDU factorization § Uniqueness § LU factorization of non-square matrices A little detour into Chapter 2 that we should know @2019 Arizona State University.
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS Consider system of equations x + 2 y + z = 2 3 x + 8 y + z = 12 4 y + z = 2 Put it in a matrix form x y z = 2 12 2
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS Start row eliminations of A § How can we describe this operation with the help of a matrix multiplication? § I.e. what would be the matrix E 21 , such that E 21 A gives us the matrix on the right?
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS Start row eliminations of A E 21 A
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS Start row eliminations of A § OK, we are finished with the first pivot: it has only zeroes beneath it. § We got lucky: we only had to perform E 21 operation (worked with 2 nd row, 1 st column) § We did not have to perform E 31 operation, because there is already zero there, otherwise we would!
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS In this case, for consistency, we can say that E 31 is the identity matrix E 31 =
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS OK, now the second pivot § Since it is a 3x3 matrix, after working with the second pivot, we already got U ! § What is E 32 here?
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS OK, now the second pivot E 32 E 21 A= E 21 A E 32 U
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS E 32 E 21 A= E 21 A E 32 U E ij is called an elementary matrix that subtracts l times row j from row i . It is the identity matrix with a l in row i , column j .
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS Therefore, elimination in a matrix form is E 32 E 31 E 21 A=U § Where E ij before A are all the row operations we had to perform. § For bigger matrices, of course, there are many more operations and many more contributing E ij !
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@2019 Arizona State University. LU FACTORIZATION: ROW OPERATIONS How do we inverse elimination step, i.e. what is E’ ij such that E’ ij E ij A=A, or in other words E’ ij E ij =I?
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  • Fall '19
  • Yulia Peet

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