EEL5173 Fall 2009 Lecture _21

# EEL5173 Fall 2009 Lecture _21 - Theorem Given a RMFD Using...

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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 Using...

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