Upper and Lower Triangular Matrixes
Extracs the pper or lower tridiagonal part from a matrix.
triang(x) Triang(x)
x |
a numeric matrix. |
The functions triang and Triang allow to transform a
square matrix to a lower or upper triangular form.
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 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.
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.
## Create Pascal Matrix: P = pascal(3) P ## Create lower triangle matrix L = triang(P) L
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