Transcription of 11. LU Decomposition - University of California, Davis
{{id}} {{{paragraph}}}
Example: marketing
11. LU Decomposition - University of California, Davis
11. LU DecompositionCertain matrices are easier to work with than others. In this section, wewill see how to write any square matrixMas the product of two matricesthat are easier to work with. We ll writeM=LU, where: Lislower triangular. This means that all entries above the maindiagonal are zero. In notation,L= (lij) withlij= 0 for allj > Uisupper triangular. This means that all entries below the maindiagonal are zero. In notation,U= (uij) withuij= 0 for allj < M=LUis called is a useful trick for many computational reasons. It is much easierto compute the inverse of an upper or lower triangular matrix. Since inversesare useful for solving linear systems, this makes solving any linear systemassociated to the matrix much faster as well. We haven t talked aboutdeterminants yet, but suffice it to say that they are important and very easyto compute for triangular systems associated to triangular matrices are very easy tosolve by back substitution.
Example Linear systems associated to triangular matrices are very easy to solve by back substitution. a b 1 0 c e!)y= e c:x= a (1 be) 0 B @ 1 0 0 d a 1 0 e b c 1 f 1 C A)x= d: y= e ad; z= f bd c(e ad) For lower triangular matrices, back substitution gives a quick solution; for upper triangular matrices, forward substitution gives the solution. 1
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: