Upper and lower triangular matrices

Extracs the upper or lower triangular part from a matrix.

Usage

triang(x)
Triang(x)

Arguments

x

a numeric matrix.

Details

triang and Triang transform a square matrix to a lower or upper triangular form. The functions just replace the remaining values with zeroes and work with non-square matrices, as well.

A triangular matrix is either an upper triangular matrix or lower triangular matrix. For the first case all matrix elements a[i,j] of matrix A are zero for i>j, whereas in the second case we have just the opposite situation. A lower triangular matrix is sometimes also called left triangular.

In fact, triangular matrices are so useful that much of computational linear algebra begins with factoring or decomposing a general matrix or matrices into triangular form. Some matrix factorization methods are the Cholesky factorization and the LU-factorization. Even including the factorization step, enough later operations are typically avoided to yield an overall time savings.

Triangular matrices have the following properties: the inverse of a triangular matrix is a triangular matrix, the product of two triangular matrices is a triangular matrix, the determinant of a triangular matrix is the product of the diagonal elements, the eigenvalues of a triangular matrix are the diagonal elements.

Value

a matrix of the same dimensions as x with the elements above or below the main diagonal set to zeroes

References

Higham, N.J., (2002); Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM.

Golub, van Loan, (1996); Matrix Computations, 3rd edition. Johns Hopkins University Press.

Examples

## Create Pascal Matrix:
P = pascal(3)
P
#>      [,1] [,2] [,3]
#> [1,]    1    1    1
#> [2,]    1    2    3
#> [3,]    1    3    6
   
## Create lower triangle matrix
L = triang(P)
L                                
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    1    2    0
#> [3,]    1    3    6