Search results
Results from the WOW.Com Content Network
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
Lewy's example takes this latter equation and in a sense translates its non-solvability to every point of . The method of proof uses a Baire category argument, so in a certain precise sense almost all equations of this form are unsolvable. Mizohata (1962) later found that the even simpler equation
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations. Classical elimination theory culminated with the work of Francis Macaulay on multivariate resultants, as ...
For a matrix with integer coefficients, the Hermite normal form is a row echelon form that can be calculated without introducing any denominator, by using Euclidean division or Bézout's identity. The reduced echelon form of a matrix with integer entries generally contains non-integer entries, because of the need of dividing by its leading ...
Second, we solve the equation = for x. In both cases we are dealing with triangular matrices (L and U), which can be solved directly by forward and backward substitution without using the Gaussian elimination process (however we do need this process or equivalent to compute the LU decomposition itself).
Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form .
In other situations, the system of equations may be block tridiagonal (see block matrix), with smaller submatrices arranged as the individual elements in the above matrix system (e.g., the 2D Poisson problem). Simplified forms of Gaussian elimination have been developed for these situations. [6]
Because of this, often, the only simple effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation. After discarding solutions that yield an invalid equation, we will have the correct set of solutions.