Linear Algebraic Equations

Transcription

1 Lier lgeric Equtios Phildelphi Uiversity Egieerig Fculty Mechicl Egieerig Deprtmet Eg Lith trseh Mtrices mtri ottio m m

2 Specil cses o Mtrices Vector mtri m Squre mtri (=m) m m m mm Specil cses o Mtrices Squre mtri Symmetric mtri ( ij = ji ) = = = digol mtri ij = ; i j ij ; i = j

3 Specil cses o Mtrices Squre mtri Idetity mtri ij = ; i j ij ; i = j Upper d lower trigulr mtrices Upper mtri Lower mtri Specil cses o Mtrices Squre mtri ouded mtri This emple hs dwidth equl

4 Mtrices opertios dditio [C] = []+[] the c ij = ij + ij []+[] = [] +[] ([]+[]) +[C] = [] +([] + [C] Multiplictio ([] []) [C] = [] ([] [C]) [] ([]+[C])=[][]+[][C] [][] [][] uless [][] - =[] - []=[I] Multiplictio [D] = g[]+[] g g g g [C] = [] [] ; c g g g c ih g g g ik k kj Mtrices opertios Mtri Determet det -

5 Mtrices opertios Mtri trspose d iverse Trspose : [] ij = [] ji [ ] Iverse [] - = [ ] det [ ] T [ coctor mtri o ] T Mtrices opertios Mtri iverse djoit method [ ] det [ coctor mtri o ] T [ ] [coctor mtri o ] [ coctor mtri o ], d so o 5

6 6 Termiology Trsormtio to mtri ottio the give equtios re i orm o d trserred to Solutio methods Grphicl method emple : Solutio Solutio C ot e pplied or more th three equtios

7 7 Solutio methods Crmer s rule Crmer s rule Emple Eq Eq Eq

8 8 Solutio methods Elimitio o ukows Forwrd elimitio Sutrct () rom (): d so: ckwrd sustitutio: * * d Solutio methods Nive Guss Elimitio Forwrd Elimitio ckwrd sustitutio " ' " ' ' ' ' ' " " / / /

9 Nive Guss Elimitio Emple Eq Eq Eq Multiply Eq y() d sutrct Eq rom it Sutrct Eq rom Eq 8 Multiply Eq y() d sutrct it rom Eq ck sustitutio / 5 ( )( 5 ) / ( )( ) ()( 5 ) / Veriictio Guss Jord Method Methodology this is doe y : Elimitig rom Eqs & Elimitig rom Eqs & Elimitig rom Eqs & 9

10 Guss Jord Method Emple Solutio Eq Eq Eq LU- Decompositio method Methodology U L D L D D U Decompositio Forwrd ckwrd Sustitutio

11 LU- Decompositio Decompositio phse Decompositio U L ' ' '' ' ' LU- Decompositio Sustitutio phse U L D D D

12 LU- Decompositio Detiled procedures decompositio phse to id [U]: multiply Eq y the ctor ( / ) Sutrct Eq rom Eq to elimite rom Eq multiply Eq y the ctor ( / ) Sutrct Eq rom Eq to elimite rom Eq t this poit, you elimite rom Eqs d d the mtri will e i the ollowig orm: Sve the vlues o d to use it lter ' ' ' ' LU- Decompositio Detiled procedures decompositio phse cotiuo to id [U]: multiply Eq y the ctor ( / ) Sutrct Eq rom Eq to elimite rom Eq t this poit, you elimite rom Eq d the mtri will e i the ollowig orm: This mtri is clled the upper mtri ' ' ''

13 LU- Decompositio Detiled procedures decompositio phse to id [L]: L ' ' ' ' ' ' LU- Decompositio Detiled procedures orwrd sustitutio phse rrge the mtrices to e like: ckwrd sustitutio phse rrge the mtrices to e like: ' L D d d d d d d d d d ' '' d d d U d D d d ' '' '

14 LU- Decompositio Emple , U 7 L Decompositio phse 7 U L Veriictio LU- Decompositio Emple Sustitutio phse d d d d d d D L D U Forwrd sustitutio ckwrd sustitutio