XV. On systems of linear indeterminate equations and congruences

  title={XV. On systems of linear indeterminate equations and congruences},
  author={Henry John Stephen Smith},
  journal={Philosophical Transactions of the Royal Society of London},
  pages={293 - 326}
  • H. Smith
  • Mathematics
  • Philosophical Transactions of the Royal Society of London
The theory of the solution, in positive or negative integral numbers, of systems of linear indeterminate equations, requires the consideration of rectangular matrices, the constituents of which are integral numbers. It will therefore be convenient to explain the meaning which we shall attach to certain phrases and symbols relating to such matrices. A matrix containing p constituents in every horizontal row, and q in every vertical column, is a matrix of the type q × p. We shall employ the… 
On Computing the Smith Normal Form of an Integer Matrix
The invariant factor theorem is then referred to as Smith 's Theorem and was originally proved in 1861 [12].
Solving Systems of Linear Equations over Polynomials
  • R. Kannan
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1985
Applications of Polynomial Smith Normal form Calculations
An integer Square matrix with a deteminant of + 1 or −1 is called unimodular. Given a (m,m) integer matrix A, there exist unimodular matrices U,K such that S(A)=UAK is a diagonal matrix with positive
Weierstrass and the theory of matrices
The following essay is the third in a series devoted to the history of the theory of matrices.1 In [1975 a] I related Cauchy's important memoir on the characteristic roots of a real quadratic form to
Algebraic Multiplicity of Eigenvalues of Linear Operators
This book brings together all the most important known results of research into the theory of algebraic multiplicities, from well-known classics like the Jordan Theorem to recent developments such as
A solution to the extended GCD problem with applications
The g,cd algorithm presented here has numerou supplications and is already led tofaster algorithms for computing row reduced echelon forms of integer matrices and solving systems of linear Diophantine equations.
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
  • J. M. Rojas
  • Mathematics, Computer Science
    J. Complex.
  • 2000
An algorithm for approximating the real roots of certain sparse polynomial systems with simple and efficient generalization to certain univariate exponential sums and a new and sharper upper bound on the number of connected components of a semi-algebraic set are given.