Elementary Row Operations

The elementary row operation rowadd adds multiples of one or more rows to other rows of a matrix. This is usually used as a means to solve systems of linear equations, of the form \(A x = b\), and rowadd corresponds to adding equals to equals.

Usage

rowadd(x, from, to, mult)

Arguments

x: a numeric matrix, possibly consisting of the coefficient matrix, A, joined with a vector of constants, b.
from: the index of one or more source rows. If from is a vector, it must have the same length as to.
to: the index of one or more destination rows
mult: the multiplier(s)

Value

the matrix x, as modified

Details

The functions rowmult and rowswap complete the basic operations used in reduction to row echelon form and Gaussian elimination. These functions are used for demonstration purposes.

Examples

A <- matrix(c(2, 1, -1,
             -3, -1, 2,
             -2,  1, 2), 3, 3, byrow=TRUE)
b <- c(8, -11, -3)

# using row operations to reduce below diagonal to 0
Ab <- cbind(A, b)
(Ab <- rowadd(Ab, 1, 2, 3/2))  # row 2 <- row 2 + 3/2 row 1
#>                   b
#> [1,]  2 1.0 -1.0  8
#> [2,]  0 0.5  0.5  1
#> [3,] -2 1.0  2.0 -3
(Ab <- rowadd(Ab, 1, 3, 1))    # row 3 <- row 3 + 1 row 1
#>                 b
#> [1,] 2 1.0 -1.0 8
#> [2,] 0 0.5  0.5 1
#> [3,] 0 2.0  1.0 5
(Ab <- rowadd(Ab, 2, 3, -4))   # row 3 <- row 3 - 4 row 2
#>                 b
#> [1,] 2 1.0 -1.0 8
#> [2,] 0 0.5  0.5 1
#> [3,] 0 0.0 -1.0 1
# multiply to make diagonals = 1
(Ab <- rowmult(Ab, 1:3, c(1/2, 2, -1)))
#>                  b
#> [1,] 1 0.5 -0.5  4
#> [2,] 0 1.0  1.0  2
#> [3,] 0 0.0  1.0 -1
# The matrix is now in triangular form

# Could continue to reduce above diagonal to zero
echelon(A, b, verbose=TRUE, fractions=TRUE)
#> 
#> 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  

# convenient use of pipes
I <- diag( 3 )
AA <- I |>
  rowadd(3, 1, 1) |>   # add 1 x row 3 to row 1
  rowadd(1, 3, 1) |>   # add 1 x row 1 to row 3
  rowmult(2, 2)        # multiply row 2 by 2

Usage

Arguments

Value

Details

See also

Examples