Mathematical Foundations of Software Development: by Hartmut Ehrig (auth.), Hartmut Ehrig, Christiane Floyd,

By Hartmut Ehrig (auth.), Hartmut Ehrig, Christiane Floyd, Maurice Nivat, James Thatcher (eds.)

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Extra info for Mathematical Foundations of Software Development: Proceedings of the International Joint Conference on Theory and Practice of Software Development (TAPSOFT) Berlin, March 25–29, 1985 Volume 1: Colloquium on Trees in Algebra and Programming (CAAP' 85)

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M , where qi1(s) is the quotient and ri1(s) the remainder of division of the polynomial a i1(s) by a 11(s). Carrying out L[i+1u(-qi1(s))], we replace the entry a i1(s) with the remainder ri1(s). If not all remainders are equal to zero, then we choose this one, that is the polynomial of the lowest degree, and carrying out operations L[i, j], we move it to position (1,1). Denoting this remainder by r i1(s), we repeat the above procedure taking the remainder r 11(s) instead of a 11(s). The degree r 11(s) is lower than the degree of a 11(s).

1. Carrying out only elementary operations on rows or columns it is possible to transform a nonsingular polynomial matrix to one of column reduced form and row reduced form, respectively. 8 Reduction of Polynomial Matrices to the Smith Canonical Form mun Consider a polynomial matrix A(s) [s] of rank r. 1. A polynomial matrix of the form A S (s) 0 ªi1 ( s) « 0 i (s) 2 « « # # « 0 « 0 « 0 0 « # « # « 0 0 ¬ ! % ! % ! 0 0 ! 0º 0 0 ! 0 »» # # % #» » ir ( s ) 0 ! 0 »   mun [ s ] . 0 0 ! 0» » # # % #» 0 0 !

1). ( s  sq ) mkq . 1) From divisibility of the polynomial ik+1(s) by the polynomial ik(s) it follows that mr ,1 t mr 1,1 t ... t m1,1 t 0 . mr ,q t mr 1,q t ... 2) If, for example, i1(s) = 1, then m11 = m12 = … =m 1q = 0. 1. 1) is called elementary divisor of the matrix A(s). 3) are (s+2) and (s+2, 5). The elementary divisors of a polynomial matrix are uniquely determined. This follows immediately from the uniqueness of the invariant polynomial of polynomial matrices. Equivalent polynomial matrices possess the same elementary divisors.

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