MATH 416 L U Decomposition FEB 19LAST TIME I Given any A CMmm F Here is a B CMmm F

in RR EF elementary mature E EntMmIIF s.t A E I B1 A invertible RR EFCA ImIn that case RR EF A i Im In A

THISTIME The LU decomposition

Here L stands for a lowertriangular meth xU for an upper triangular meths

A matrix A cMmmCFI is lower triangularif ai o whenever is

et II IA matrix AEMm.nl lisupperthansul r ea Laoif aij 0 whenever i j O O

E g The elementary matrices Pgi is lower triangular if icjUpper triangular if i j

For example Ps s L Ps L's II

If we're given a matrix A I and we wanted to use row

opertio s to make it upper triangular ie to get d g h O

then in the simplest case we would only need to use row

operation 131 repeatedly and add a multiple of a row to a

lower row

In that case we would have Pc in Pa ii i A U ie Je Hl

Solving for A we find A Pc in Pai A P a i j Pcn.in UL

Algorithm To find the LU decomposition of a matrix A cMmmCFi Try to reduce A to an uppertriangular matrix using only

row opertions RI only ever adding a multiple of ro w j to ro i

if i 2 j

If this succeed call the result UIf it fails ten A does not have an LU decomposition

ii If the previous step succeeded define L CMmlF to be thelower triangular matrix 41 s on the diagonal

c in the Li j entry if during the elimination process

a times now j was added to now i

Then A LU

E g Find the LU decomposition if it exists of A Ig

i i n i iU P4,4 a P z s aP l u iD5,3 i B r i F

Hence a E a

1 L i ItGiven A LU we solve Ax b LUX b

by setting y Ux s that Ly b

It is easy to solve L y b once we know yten it is easy to solve Ux y to find x

This is usually the preferred methodfor solving linear systems

E g with A a above let's sole Ax

First we find the unique solution to L y b which

mists 4 44 1

Next we solve Ux y with Xs as free variable

i Xy I 5 5

and so the set of solutions is Ig tf te IR

Sometime a matrix does not have an LU decompositionfor ex me.fi file L Liii e ii iii I

we wouldneed l U O l U I lie

If we allow ourselves to switch to order ofthe rows firstthen we can alway find an LU decomposition

The For every ACMm u F there is a matrix P eMmHTT

which is a product of elementary metrics oftheform Ri jan uppertriangular matrix UtMm n IIF a lower triangular Lemon F

such that PA LU

This is the LU 4pivoting decomposition of A

PI Any A can be put into uppertriangular form by usingrow operations of type I 3 so all we need to check is

that it is possible to carry out the new exchanges first then

the row operations of type 1This follows from Ri Pc he Pak e Ri if i j3A k l 4

Pc eRi i if i k it l

if i l jfk