EEL5173 Fall 2009 Lecture _21

EEL5173 Fall 2009 Lecture _21 - Theorem. Given a RMFD,...

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Unformatted text preview: Theorem. Given a RMFD, Using elementary row operations, reduce the matrix H into the Column Hermite Form is co-prime. ), ( ) ( ) ( 1 s Q s P s G R R = ( )( ) 1 1 1 ) ( = R R R R H Q H P s G let = R R P Q s H ) ( Consequently, ) ( R H s H H R is then a gcrd of P R and Q R . Note: Since Q R in invertible, rank{ H } = q . Hence H R is also of rank q and therefore invertible. p x q p x q q x q p x q q x q v. Co-prime RMFD via gcrd (continue) Ex. 1 2 2 2 2 1 2 2 2 2 2 ) 2 ( ) 2 ( ) 1 ( ) 1 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 1 ( ) ( + + + + = = + + + + + = s s s s s s s s Q P s s s s s s s s s s G R R Find a gcrd of P R and Q R and then a co-prime RMFD of G ( s ). + + + + + + + + = = s s s s s s s s s s s s s s s s P Q s H R R 2 2 2 2 2 2 2 2 ) 1 ( ) 2 ( ) 1 ( ) 2 ( ) 1 ( ) 2 ( ) 2 ( ) 1 ( ) ( ) 4 3 )( 3 ( 4 ) 2 ( ) 1 ( ) ( ) 2 ( 4 ) ( 1 ) 2 ( ) 2 ( ) ( 4 ) 2 ( ) 2 ( ) ( ) ( 4 ) ( ) 2 ( ) 2 ( 4 ) ( ) 2 ( ) 1 ( ) 1 ( ) 1 ( 4 4 ) 2 ( ) 1 ( ) 2 ( ) 1 ( ) 2 ( ) 1 ( ) 2 ( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + = + + =...
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EEL5173 Fall 2009 Lecture _21 - Theorem. Given a RMFD,...

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