Echelon Form of a Matrix

Source: R/gaussian-elimination.R

echelon.Rd

Returns the (reduced) row-echelon form of the matrix A, using gaussianElimination .

Usage

echelon(A, B, reduced = TRUE, ...)

Arguments

A: coefficient matrix
B: right-hand side vector or matrix. If B is a matrix, the result gives solutions for each column as the right-hand side of the equations with coefficients in A.
reduced: logical; should reduced row echelon form be returned? If FALSE a non-reduced row echelon form will be returned
...: other arguments passed to gaussianElimination

Value

the reduced echelon form of X.

Details

When the matrix A is square and non-singular, the reduced row-echelon result will be the identity matrix, while the row-echelon from will be an upper triangle matrix. Otherwise, the result will have some all-zero rows, and the rank of the matrix is the number of not all-zero rows.

Author

John Fox

Examples

A <- matrix (c (2, 1, -1,
 -3, -1, 2,
 -2, 1, 2), 3, 3, byrow=TRUE)
b <- c (8, -11, -3)
echelon(A, b, verbose=TRUE, fractions=TRUE) # reduced row-echelon form
#> 
#> Initial matrix:
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 2 1 -1 8 
#> [2,] -3 -1 2 -11 
#> [3,] -2 1 2 -3 
#> 
#> row: 1 
#> 
#> exchange rows 1 and 2 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] -3 -1 2 -11 
#> [2,] 2 1 -1 8 
#> [3,] -2 1 2 -3 
#> 
#> multiply row 1 by -1/3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 2 1 -1 8
#> [3,] -2 1 2 -3
#> 
#> multiply row 1 by 2 and subtract from row 2 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1/3 1/3 2/3
#> [3,] -2 1 2 -3
#> 
#> multiply row 1 by 2 and add to row 3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1/3 1/3 2/3
#> [3,] 0 5/3 2/3 13/3
#> 
#> row: 2 
#> 
#> exchange rows 2 and 3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 5/3 2/3 13/3
#> [3,] 0 1/3 1/3 2/3
#> 
#> multiply row 2 by 3/5 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1 2/5 13/5
#> [3,] 0 1/3 1/3 2/3
#> 
#> multiply row 2 by 1/3 and subtract from row 1 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 -4/5 14/5
#> [2,] 0 1 2/5 13/5
#> [3,] 0 1/3 1/3 2/3
#> 
#> multiply row 2 by 1/3 and subtract from row 3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 -4/5 14/5
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1/5 -1/5
#> 
#> row: 3 
#> 
#> multiply row 3 by 5 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 -4/5 14/5
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1 -1
#> 
#> multiply row 3 by 4/5 and add to row 1 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 2
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1 -1
#> 
#> multiply row 3 by 2/5 and subtract from row 2 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 2 
#> [2,] 0 1 0 3 
#> [3,] 0 0 1 -1 
echelon(A, b, reduced=FALSE, verbose=TRUE, fractions=TRUE) # row-echelon form
#> 
#> Initial matrix:
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 2 1 -1 8 
#> [2,] -3 -1 2 -11 
#> [3,] -2 1 2 -3 
#> 
#> row: 1 
#> 
#> exchange rows 1 and 2 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] -3 -1 2 -11 
#> [2,] 2 1 -1 8 
#> [3,] -2 1 2 -3 
#> 
#> multiply row 1 by -1/3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 2 1 -1 8
#> [3,] -2 1 2 -3
#> 
#> multiply row 1 by 2 and subtract from row 2 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1/3 1/3 2/3
#> [3,] -2 1 2 -3
#> 
#> multiply row 1 by 2 and add to row 3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1/3 1/3 2/3
#> [3,] 0 5/3 2/3 13/3
#> 
#> row: 2 
#> 
#> exchange rows 2 and 3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 5/3 2/3 13/3
#> [3,] 0 1/3 1/3 2/3
#> 
#> multiply row 2 by 3/5 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1 2/5 13/5
#> [3,] 0 1/3 1/3 2/3
#> 
#> multiply row 2 by 1/3 and subtract from row 3 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1/5 -1/5
#> 
#> row: 3 
#> 
#> multiply row 3 by 5 
#> Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent approaches
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1 -1

A <- matrix (c (1,2,3,4,5,6,7,8,10), 3, 3) # a nonsingular matrix
A
#> [,1] [,2] [,3]
#> [1,] 1 4 7
#> [2,] 2 5 8
#> [3,] 3 6 10
echelon(A, reduced=FALSE) # the row-echelon form of A
#> [,1] [,2] [,3]
#> [1,] 1 2 3.333333
#> [2,] 0 1 1.833333
#> [3,] 0 0 1.000000
echelon(A) # the reduced row-echelon form of A
#> [,1] [,2] [,3]
#> [1,] 1 0 0
#> [2,] 0 1 0
#> [3,] 0 0 1

b <- 1:3
echelon(A, b) # solving the matrix equation Ax = b
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 1
#> [2,] 0 1 0 0
#> [3,] 0 0 1 0
echelon(A, diag (3)) # inverting A
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 0 0 -0.6666667 -0.6666667 1
#> [2,] 0 1 0 -1.3333333 3.6666667 -2
#> [3,] 0 0 1 1.0000000 -2.0000000 1

B <- matrix (1:9, 3, 3) # a singular matrix
B
#> [,1] [,2] [,3]
#> [1,] 1 4 7
#> [2,] 2 5 8
#> [3,] 3 6 9
echelon(B)
#> [,1] [,2] [,3]
#> [1,] 1 0 -1
#> [2,] 0 1 2
#> [3,] 0 0 0
echelon(B, reduced=FALSE)
#> [,1] [,2] [,3]
#> [1,] 1 2 3
#> [2,] 0 1 2
#> [3,] 0 0 0
echelon(B, b)
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 -1 1
#> [2,] 0 1 2 0
#> [3,] 0 0 0 0
echelon(B, diag (3))
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 0 -1 -1.0 0 0.6666667
#> [2,] 0 1 2 0.5 0 -0.1666667
#> [3,] 0 0 0 -0.5 1 -0.5000000