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EEL5173 Fall 2009 Lecture _19

# EEL5173 Fall 2009 Lecture _19 - 4 Co-prime MFD irreducible...

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4. Co-prime MFD irreducible? for an si-so system, no common root between the numerator and the denominator Ex. [ ] [ ] . ), 1 ( 5 ) 2 ( ) 2 ( ) 1 ( 5 ) 2 ( ) 1 ( 5 ) ( 1 2 1 2 2 e irreducibl s s s s s s s G + + = + + = + + = Characterization of the common factors (divisors)? Any common factor (divisor) between the numerator and the denominator must be a non-zero constant. Generalization?

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i. Unimodular Matrix Ex. { } , 1 1 0 1 1 1 0 1 1 0 1 1 ) ( ) ( ) ( 1 0 1 ) ( 1 = = = = = s s s s s s s L s L adj s L s s L T Definition . Let L ( s ) be a non-singular square matrix of polynomials. The matrix L ( s ) is said to be unimodular if its inverse is a matrix of polynomials. not unimodular.
Any non-singular square matrix of constants is unimodular. Any others? Ex. { } , 1 0 ) 3 ( 1 1 ) 3 ( 0 1 1 1 ) ( ) ( ) ( 1 0 ) 3 ( 1 ) ( 2 2 1 2 = = = = s s s L s L adj s L s s L T unimodular. Theorem . A non-singular square matrix of polynomials is unimodular if and only if its determinant is a non- zero constant.

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