206 - Suppose an mn matrix A can be transformed into a row...

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1 Suppose an m × n matrix A can be transformed into a row echelon form U only with the elementary row operations of adding multiples of some rows to others. Let U = E k E k 1 E 1 A , where E p is the p th elementary matrix corresponding the p th elementary row operation. Then where 111 21 Example: 1 0 001 0 0010 0 0 1 0 0 0 0 1 000 1 0001 0 0 0 1 0 0 . 0 100 0 1000 1 0 0 1 0 00010 0100 1 kkk EEE aa bc b c −−− −− ⎡⎤ ⎢⎥ = ⎣⎦ ±²²³²²´±²²³²²´±²²³²²´ L = [ l ij ] is a unit ( l ii = 1 for i = 1, …, m ) lower triangular matrix ( l ij = 0 for i < j ), and l ij = c ij for i > j if in an elementary row operation c ij times of row j is added to row i . U = [ u ij ] is an upper triangular matrix ( u ij = 0 for i > j ) because it is in a row echelon form. A = LU is the LU decomposition of A .
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2 Example Suppose a matrix A has an LU decomposition A = LU . The system of linear equations A x = b can be re-written as A x = LU x = L ( U x ) = b , and can be solved by 1) solving L y = b first and 2) doing the back substitution U x = y .
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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206 - Suppose an mn matrix A can be transformed into a row...

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