Lecture 26 - Mar 11

Lecture 26 - Mar 11 - Wednesday March 11 Lecture 26 :...

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Wednesday March 11 Lecture 26 : Matrix factorization (Refers to 3.5) Expectations : 1. Define the diagonal, upper triangular, lower triangular and triangular for an m × n matrix A . 2. Define the LU -factorization of an m × n matrix A . 26.1 Definition If A is an m × n matrix [ a ij ] m × n we will call the diagonal of A those entries a 11 , a 22 , a 33 , . .., a mm . Notice that we are defining the diagonal of a matrix which is not necessarily square. An m × n matrix A where all entries above the main diagonal are zeroes is called a lower triangular m × n matrix. That is if j > i , then a ij = 0. An upper triangular m × n matrix is one where all entries below the main diagonal are zeroes. That is if i > j , then a ij = 0. A triangular m × n matrix A is one which is either upper or lower triangular. 26.1.1 Examples of upper triangular matrices: 1 1 03 02105 111 0 211 00031 0 1 1 00 3 0 00101 000 26.1.2 Remark Triangular matrices are of interest since systems A x = b where the coefficient matrix A is triangular are easy to solve. An example is given. x 1 + 4 x 2 3 x 3 x 4 + 5 x 5 = 4 5 x 3 + x 4 + x 5 = 8 2 x 5 = 6
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This system is easily seen to have the solution ( x 1 , x 2 , x 3 , x 4 , x 5 ) = ( 9 2 x 2 + (2/5) x 4 , x 2 , 1 (1/5) x 4 , x 4 , 3) 26.2 Lemma ( Optional ) The product of two upper triangular matrices is again upper triangular. The product of two lower triangular matrices is again lower triangular.
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This note was uploaded on 06/10/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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Lecture 26 - Mar 11 - Wednesday March 11 Lecture 26 :...

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