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2 Solving linear systems
we're interested iu solving the linear system
Ax b AE IR BE IN MEN
I itRecall's A e IR
h AA A A Iinverse exists if detCA 0
we call these matrices nonsing a Fae or
Cmatrix ainvertible
Gamel'srule det Aib with i th columnreplacedby b
diet A
requires htt determinants which isexpensive computationally
Computing the determinant requiresn operations
ie additions summationsfloatingpoint operations flops
2 Gaussian elimination
example III H 1 EEGenerate triangularsystem by adding multiples ofrows to other rows this does not change the solution
addC2 firstnow
us III of xto3rdrow
this is identical to multiplying the systemwith
4 12 9 and Lz I 9 fromtheleft
Ita E
c sixsecond2 E go
I us E
IsamuLig g's fly
IfTuppertriangularmatrix
A can be solved by backwards substitution
4 La downtriangular x 4matrices
Defi LE IR lowutriangulai if Lig O for allios
unitlowutriangulae if additionallyhi tha
Analogue for upper triangular unit uppa triangularmatrices
Thmi properties of lower triangular matrices identical resultsholdfor upper triangular
i products of lower triangularmollies
matrices are lowee triangularii as above for unit lower triangulariii lower triangular matrices are non singulee if
l 1 0 lnu F O
N invertible lowee triangular matrices have lower triangularinverses
v same as Gv for unit lower triangular
Proefof rest is easy Induction over matrix size n
n Abe E defE I D a e O a to be o
ni HSince L If IRM
xChH
D L L I e IRh L c O RTL tart 0
Li Li is low triangular rEtqu I
due to induction assumption
L c O D C O D L is howa triangularB