MATRIX AND VECTOR NORMS

Transcription

1 Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the solute vlue tells how lrge umer s. VECTOR NORMS Vectors re secl kd of mtrces. vector s sgle colum mtr. From eerece, two-dmesol le (D) or three-dmesol sce (3D), the legth of vector s ecellet mesure of the sze vector. Cosder the 3D vector [, y, z] T, the legth of the vector s defed s y z The legth of the vector s geerlly greter th zero. The legth s zero oly for the zero vector [0, 0, 0] T. Rule 0, oly for the zero vector. Whe vector s multled y costt k ( ecomes k [ k, ky, kz] T ), the legth of the vector k k z k y k stes.google.com/ste/zydmsoud/umercl 5

2 Numercl lyss for Egeers Germ Jord Uversty k k y z k Rule k k Cosder the vector d show c However, the legth of the vector c s less the the sum of legths of the vectors d, d equls the sum oly whe vectors d re le. e., c c Rule 3 The mgtude of the cross-roduct of the vectors d s s However, the mgtude of s s less th or equl to. The Rule stes.google.com/ste/zydmsoud/umercl 6

3 Numercl lyss for Egeers Germ Jord Uversty The ove four rules c e used s geerl rules for defg orm of vector or mtr. These rules c e geerlzed s follows: For mtrces d B, the orm must stsfy the followg:. Norm ( ) 0 Norm ( ) 0 oly for the zero mtr.. Norm ( k) k. Norm ( ) where k s costt. 3. Norm ( B) Norm ( ) Norm ( B). Norm ( B) Norm ( ). Norm ( B) The orm of vector or mtr s deoted y the symol. geerl defto of vector orm tht stsfes ll ove codtos s The most commoly used orm re Sum of mgtudes orm ( ) Euclde orm ( ) e stes.google.com/ste/zydmsoud/umercl

4 Numercl lyss for Egeers Germ Jord Uversty Ifty orm or mmum mgtude orm ( ) lm m Emle For the vector The oe orm [.5, 0.0, 5.5] T, The Euclde orm The fty orm or mmum mgtude orm m MTRIX NORMS The orm deftos for vectors c e geerlzed for mtrces s follows: Euclde orm e j j stes.google.com/ste/zydmsoud/umercl 8

5 Numercl lyss for Egeers Germ Jord Uversty stes.google.com/ste/zydmsoud/umercl 9 Note tht for mtrces, e. The s clled the Sectrl orm d s relted to the egevlues of the mtr. Mmum colum sum (oe orm) j j m Mmum row sum (fty orm) j j m Emle Cosder the mtr Euclde orm j j e Mmum colum sum (oe orm) 6 5 m j j Mmum row sum (fty orm)

6 Numercl lyss for Egeers Germ Jord Uversty m j j MTRIX CONDITION NUMBER The mtr codto umer s defed s cod ( ) The codto umer of mtr s greter th or equl to. It c e show tht cod ( ) Ths mes tht the reltve error the orm of comuted soluto vector c e s lrge s the reltve error the orm of the mtr multled y the codto umer of the mtr. Lrger codto umer mgfes the errors the mtr whch reflects o the level of cofdece the soluto vector. Usg sgfct fgures clcultos, the smllest mesurle error s Emle For umer.65 0, the closest romto s ether.6560 or The error s ether romto s The smllest reltve error for the closest romto c e determed s of the umer 0 stes.google.com/ste/zydmsoud/umercl 30

7 Numercl lyss for Egeers Germ Jord Uversty t whch s equvlet to Emle s 0 where s s the umer of sgfct fgures used. Usg 6 sgfct fgures clcultos, f the codto umer of mtr s 000, the reltve error the orm of soluto vector c e s lrger s Emle Cosder the mtr Usg sgfct fgures clcultos The verse of the mtr s cod ( ) stes.google.com/ste/zydmsoud/umercl 3

8 Numercl lyss for Egeers Germ Jord Uversty The fty orms of the mtr d ts verse re m j m j j The codto umer of the mtr s j cod ( ) whch dctes very ll-codtoed mtr. The lless of the mtr c e verfed y multlyg the mtr wth ts verse. The multlcto result should e the detty mtr I. The frthest the multlcto result s from the detty mtr, the worst the codto of the mtr s The result of the ove multlcto oerto shows lrge devtos from the elemets of the true detty mtr I. stes.google.com/ste/zydmsoud/umercl 3