A Generalized Sylvester Identity and Fraction-free Random Gaussian Elimination

AbstractSylvester’s identity is a well-known identity that can be used to prove that certain Gaussian elimination algorithms are fraction free. In this paper we will generalize Sylvester’s identity and use it to prove that certain random Gaussian elimination algorithms are fraction free. This can be used to yield fraction free algorithms for solving Ax=b(x≥ 0) and for the simplex method in linear programming

A generalized Sylvester identity and fraction-free random Gaussian elimination

Sylvester's identity is a well-known identity which can be used to prove that certain Gaussian elimination algorithms are fraction-free. In this paper we will generalize Sylvester's identity and use it to prove that certain random Gaussian elimination algorithms are fraction-free. This can be used to yield fraction-free algorithms for solving Ax = b (x 0) and for the simplex method in linear programming

A generalized Sylvester identity and fraction-free random Gaussian elimination

Sylvester's identity is a well-known identity which can be used to prove that certain Gaussian elimination algorithms are fraction-free. In this paper we will generalize Sylvester's identity and use it to prove that certain random Gaussian elimination algorithms are fraction-free. This can be used to yield fraction-free algorithms for solving Ax = b (x 0) and for the simplex method in linear programming. 1 Introduction Sylvester's identity is a well-known identity relating a hyperdeterminant of a matrix (i.e. a determinant of minors) to the determinant of that matrix. Let R be a commutative ring and A = (a ij ) an n \Theta m matrix over R. For 0 k ! min(n; m), k ! i n and k ! j m define a (k) i;j = fi fi fi fi fi fi fi fi fi a 11 \Delta \Delta \Delta a 1k a 1j . . . . . . . . . a k1 \Delta \Delta \Delta a kk a kj a i1 \Delta \Delta \Delta a ik a ij fi fi fi fi fi fi fi fi fi : (1) We can now state Sylvester's identity (for a proof see for example [1]). Theorem 1 (Sylvest..