3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS
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1 . REVIEW OF PROPERTIES OF EIGENVLUES ND EIGENVECTORS. EIGENVLUES ND EIGENVECTORS We hll ow revew ome bc fct from mtr theory. Let be mtr. clr clled egevlue of f there et ozero vector uch tht Emle: Let Coder. We hve ( Hece uch tht there et ozero vector uch tht. Thu egevlue of. Smlrly, f we tke, we fd tht. Thu, lo egevlue of. Let be egevlue of. The y ozero uch tht clled egevector of. Let be egevlue of. Let,
2 W { C : } The we hve the followg roerte of W : ( W oemty, ce the zero vector, (whch we deote by θ W, tht, θ θ θ. (,y W, y y ( y ( y y W, ( For y cott k, we hve k k (k (k (k k W Thu W ubce of C. Th clled the chrctertc ubce or the egeubce correodg to the egevlue. Emle: Coder the the emle o ge. We hve ee tht - egevlue of. Wht W(, the egeubce correodg to? We wt to fd ll uch tht -, tht, (I θ, tht, we wt to fd ll oluto of the homogeeou ytem M θ ; where M I 6 We ow c ue our row reducto to fd the geerl oluto of the ytem. M R R R R R
3 Thu, Thu the geerl oluto of (I θ where d re rbtrry cott. Thu cot of ll vector of the form. Note: The vector, form b for ω - d therefore dm W(. Wht W ( the egeubce correodg to the egevlue for the bove mtr? We eed to fd ll oluto of,.e., θ.e., N θ where
4 6 I N g we ue row reducto N R R R R R R d ; The geerl oluto Thu ( W cot of ll vector of the form κ Where κ rbtrry cott.
5 Note: The vector dm. W (. form b for W ( d hece Now whe c clr be egevlue of mtr of order? We hll ow vetgte th queto. Suoe egevlue of. Th There ozero vector uch tht. ( I θ d θ The ytem ( I θ h t let oe ozero oluto. ullty ( - I rk ( - I < ( - I gulr det. ( - I Thu, egevlue of det. ( - I. Coverely, clr uch tht det. ( - I. Th ( - I gulr rk ( - I < ullty ( - I The ytem ( I θ h ozero oluto. egevlue of. Thu, clr uch tht det. ( - I egevlue. Combg the two we get, egevlue of det. ( - I det. (I - Now let C( det. (I - Thu we ee tht, The egevlue of mtr re recely the root of C( det. (I -. 5
6 6 We hve, ( C ( (. det Thu ; C( olyoml of degree. Note the ledg coeffcet of C( oe d hece C( moc olyoml of degree. Th clled CHRCTERISTIC POLYNOMIL of. The root of the chrctertc olyoml re the egevlue of. The equto C( clled the chrctertc equto. Sum of the root of C( Sum of the egevlue of......, d th clled the TRCE of. Product of the root of C( Product of the egevlue of det.. I our emle ge we hve ( ( det I C 7 C C C ( 7
7 R R R R ( ( ( ( ( ( Thu the chrctertc olyoml C ( ( ( The egevlue re (reeted twce d. Sum of egevlue (- (- Trce Sum of dgol etre. Product of egevlue (- (- ( det.. Thu, f mtr, we defe the CHRCTERISTIC POLYNOMIL, C( I ( d oberve tht th moc olyoml of degree. Whe we fctorze th, C( ( ( ( ( k where,,......, k re the dtct root; thee dtct root re the dtct egevlue of d the multlcte of thee root re clled the lgebrc multlcte of thee egevlue of. Thu whe C( (, the dtct egevlue re,,......, k d the lgebrc multlcte of thee egevlue re reectvely,,,....., k. For the mtr Emle ge we hve foud the chrctertc olyoml o ge 6 C ( ( ( Thu the dtct egevlue of th mtr re - ; d d ther lgebrc multlcte re reectvely ;. If egevlue of, the ege ubce correodg to d defed W C : { } k W 7
8 The dmeo of W clled the GEOMETRIC MULTIPLICITY of the egevlue d deoted by g. g for the mtr o ge, we hve foud o ge d reectvely tht, dm W( ; d dm. W (. Thu the geometrc multlcte of the egevlue - d re reectvely g ; g. Notce tht th emle t tur out tht g ; d g. I geerl th my ot be o. It c be how tht for y mtr hvg C( (, g ; k (.e., for y egevlue of, tht, geometrc multlcty lgebrc multlcty for y egevlue. We hll ow tudy the roerte of the egevlue d egevector of mtr. We hll trt wth relmry remrk o Lgrge Iterolto olyoml : Let,, , be dtct clr, (.e., j f j. Coder, ( j j ( ( ( ( ( ( ( ( j j ( ( ( ( for,, , ( The ( re ll olyoml of degree -. Further otce tht ( ( ( ( ( Thu (re ll olyoml of degree - uch tht, ( f j j δj (5 f j
9 9 We cll thee the Lgrge Iterolto olyoml. If ( y olyoml of degree - the t c be wrtte ler combto of (, (,..., ( follow: ( ( ( ( ( ( ( L.... (6 ( ( Wth th relmry, we ow roceed to tudy the roerte of the egevlue d egevector of mtr. Let,...., k be the dtct egevlue of. Let,,..., k be egevector correodg to thee egevlue reectvely ;.e., re ozero vector uch tht,,,,k (6 From (6 t follow tht ( ( ( ( d by ducto we get m m for y teger m (7 (We terret I. Now, ( be y olyoml. We defe ( the mtr, I ( Now I ( ( by (6 (. (
10 Thu, If y egevlue of d egevector correodg to,the for y olyoml ( we hve (. ( PROPERTY I Now we hll rove tht the egevector,,,...., k correodg to the dtct egevlue,,...., k of, re lerly deedet. I order to etblh th ler deedece, we mut how tht C C C θ C C C... ( Now f ( & (5 we tke k ;, (I,,.., the we get the Lgrge Iterolto olyoml d ( j k j ( ( j j ;,,.., k (9 ( j δ j ( Now, C C... θ C k k For k, ( [ C C... Ckk ] ( θ θ ( ( Ck ( k θ C C... (... C ( θ, C ( C k k k (by roerty I o ge 6 C C θ; k; by ( ; k ce re ozero vector Thu 9
11 C C... Ck k θ C C... C we hve rovg (. Thu Ege vector correodg to dtct egevlue of re lerly deedet. PROPERTY II 9
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