SECION. VECOR ND RIX NORS VECOR NOR: a fuctio from R oegatie real umbers that measures the "size" of a ector. Properties that defie a ector orm: (i) If 0, the > 0 (ii) (iii) α α for all scalars α Note: this implies that 0 0. + + for all ectors ad. he three most commo orms: i i i ma i i i Each of these is a particular case of the famil of p-orms (or Hölder orms): p i i p / p, p <. ( is obtaied b takig the limit as p ) here are a ifiite umber of was to defie a ector orm. EXPLE..5, page 5 If is a fied positie defiite matri, the is a ector orm. Proof: Verif that the three properties are satisfied. (i) Sice > 0 for all 0, > 0 if 0. 76
(ii) (iii) α + ( α) α α ( α) ( + ) + + + ( + ) + + LL + + sice sice LL But LL L, L L L b the Cauch - Schwarz iequalit (p. 3) LL LL ad thus + ( + ) ( + ) Oe wa i which orms are used: the DISNCE betwee ectors is useful whe cosiderig coergece of a sequece of ectors.. his is Sequece () () (), () () () () (), K DEFINIION lim (k ) if ad ol if lim i (k ) i for all alues i,,...,. HEORE (k ) if ad ol if lim ( ) k 0 lim for a ector orm. 77
hus, for a ector orm, the size of ( k ) is used to determie if the sequece of ectors { (k ) } is coergig to some limit ector. he choice of a orm is arbitrar -- so i this sese, all orms are equialet. Because matri products B frequetl occur i applicatios, the defiitio of a RIX NOR o square matrices i R icludes a fourth coditio: (i) > 0 if 0 (ii) α α for all scalarsα (iii) + B + B for all matrices (i) B B for all matrices ad ad B B Propert (i) is sometimes called "cosistec", but we'll also use this term i a differet wa below. EXPLE.. (page 5) he Frobeius matri orm is a F ij i j. DEFINIION Gie a fied ector orm, the matri orm iduced b subordiate matri orm) is defied to be (or its ma. 0 Proof that this defies a matri orm -- heorem..6 o page 7. 78
NOE: (see Eercise..7(b) o page 8) ma 0 ma 0 ma b defiitio b propert (ii) of o lettig a ector orm hat is, a alteratie defiitio of the iduced or subordiate matri orm is ma. DEFINIION If deotes a ector orm ad β a matri orm, the these two orms are said to be cosistet if α for all matrices ad all ectors α β α. I aalzig matri/ector computatios, we wat to use cosistet orms. HEORE..4 (page 7) ector orm ad its subordiate (iduced) matri orm are cosistet. Proof. ma 0 for a fied ector 0 for all ad GEOERICL INERPREION of subordiate matri orms For a ector orm, { : } is called the uit ball. 79
EXPLE For the -orm i R, + For the -orm i R 3, the uit ball is just the uit sphere. Cosider ow the defiitio { : } look like? ma. What does the set of ectors I R ad whe is osigular, { : } is a ellipse cetered at the origi. (I higher dimesios, it is a hperellipsoid cetered at the origi.) -- thus ma is equal to the legth of the logest ais of the ellipse -- this legth is the maimum magificatio of a ector o the uit ball 80
he geometr is similar for other ector orms. I R, the uit balls for the -orm ad the orm are as follows. If both ector ad matri orms are required i aalzig some computatio, the the best choice is usuall some ector orm ad its subordiate matri orm. Howeer, the defiitio ma is clearl usuitable for computatioal purposes. heorems such as the followig are required i order to ealuate subordiate matri orms. HEORE..9 (page 9) he matri orms subordiate to ad are ad ma j ma i i j a ij a ij Proof (for the -orm) see page 9 of the tet. NOE: the subordiate orm for is ot the Frobeius matri orm. his ca be see sice (b defiitio) the subordiate orm of the idetit matri is I I ma 0 8
but a I. ij F F i j Later we will see that maimum eigealue of maimum sigular alue of **************************************************** If α ad β are scalars ad α deotes some approimatio to the alue β, the α β is called the absolute error i α, whereas β 0 ). α β β is called the relatie error (if Similarl for ectors ad matrices: if deotes a approimatio to, ad deotes a approimatio to B, the absolute ad relatie errors are gie b, B, B B 8