DIRECT METHOD FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

DIRECT METHOD FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

In this lesson we study the solution of a Cramer system Ax = B, which means that A is ° c

invertible regular or verify ° c or det (A) 6 = 0, using a direct method. Since this is any
allowing, in the absence of errors, through a number of steps nito obtain the exact solution.
In property, this does not happen in general because of the inevitable rounding errors.

• Gauss- Jordan Elimination
• 
  In mathematics, Gaussian elimination, Gaussian elimination or Gauss-Jordan elimination, so named because Carl Friedrich Gauss and Wilhelm Jordan, are linear algebra algorithms to determine the solutions of a system of linear equations, matrices and inverse finding. Asystem of equations is solved by the Gauss when their solutions are obtained by reducing an equivalent system given in which each equation has one fewer variables than the last. When applying this process, the resulting matrix is known as "stagger."

• ALGORITHM GAUSS JORDAN
• 
  1. Go to the far left column is not zero
  2. If the first line has a zero in this column, swap it with another that does not have
  3. Then, get below zero this item forward, adding appropriate multiples of row than h e row below it
  4. Cover the top row and repeat the above process with the remaining submatrix. Repeat with the rest of the lines (at this point the array is in the form of step)
  5. Starting with the last line is not zero, move up: for each row get a 1 up front and introduce zero multiples of this sum for the rows corresponding
  
  An interesting variant of Gaussian elimination is what we call Gauss-Jordan, (due to Gauss and Wilhelm Jordan mentioned), this is to be a front for getting the steps 1 to 4 (called direct path) and the time these completed and will result in the reduced echelon form matrix.

• LU DESCOMPOSITION

ts name is derived from the English words "Lower" and "Upper", which in Spanish translates as "Bottom" and "Superior." Studying the process followed in the LU decomposition is possible to understand why this name, considering how original matrix is decomposed into two triangular

• LU decomposition involves only operations on the coefficient matrix [A], providing an efficient means to calculate the inverse matrix or solving systems of linear algebra.
  
  First you must obtain the matrix [L] and the matrix [U]..
  
  [L] is a diagonal matrix with numbers less than 1 on .the diagonal. [U] is an upper diagonal matrix on the diagonal which does not necessarily have to be number one.
  
  The first step is to break down or transform [A] [L] and [U], ie to obtain the lower triangular matrix [L] and the upper triangular matrix [U]

• INVERSE MATRIX

Is the matrix we get from chang ing rows by columns. The transpose of that represented by AT.

In mathematics, especially in linear algebra, a square matrix of order n is said to be invertible, nonsingular, nondegenerate or regular if there is another square matrix of order n, called the inverse matrix of A and represented as A-1matrices.


fuente

• Shen Kangshen et al. (ed.) (1999). Nine Chapters of the Mathematical Art, Companion and Commentary, Oxford University Press. cited byOtto Bretscher (2005).

• Linear Algebra and Its Applications, Thomson Brooks/Cole, pp. 46