Page 1 Lab 5 Elementary Matrix and Linear Algebra Spring Name Due at Final Exam Score = /25

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1 Page La 5 Eleetary Matrx ad Lear Algera Sprg 0 Nae Due at Fal Exa Sore /5. (3 pots) Go to AKT Msellaeous Matheatal Utltes page at ad uerally detere the egevalues ad egevetors of the followg atres. Fll the tale ad prtout ad attah the results fro the progra. a) 3 A det[ A ] 3 3 ) A det[ A ] (3 pots) a) Prove that the traspose of a atrx has the sae egevalues as the orgal atrx. Is the sae true of the egevetors? Expla your aswer. ) For real syetr atres the egevetors a always e expressed usg oly real uers. Is ths true of ay real square atrx? Expla your aswer. 9/4/00 Madso Area Tehal College

2 Page La 5 Eleetary Matrx ad Lear Algera Sprg 0 3. ( pots) Coplex Ier Produt Spaes, Utary ad Herta atres. The oept of a er produt a e exteded to vetors osstg of oplex etres. I aalogy wth R, the spae of desoal vetors wth oplex opoets s alled. The er or Herta produt of two T vetors s defed as u v ( u, v) u v uv. Here z desgates the oplex ougate of z. Reall that for a oplex uer z, z a+ wth a ad real, the oplex ougate s defed as z a. z zz a+ a a + 0. The oveto of oplex ougato o the rght the defto of u v s artrary ad soe authors defe the Herta produt wth oplex ougato o Furtherore, 5+ the left. What s portat s to e osstet. As a exaple of the Herta produt, let u 3 ad v, the u v ( 5+ )( 3 7) + ( 3)( ) + 4( + ) 5 3. a) Show that the Herta er produt satsfes the followg propertes. Here u, v, ad w represet vetors ad s a oplex salar. u v v u u+ v w u w + v w w u+ v w u + w v u v u v u v u v u u 0 wth equalty f ad oly f u 0. The Cauhy Shwartz equalty s stll true. For ay oplex salarα ad o-zero vetors u ad v, u v u+ αv u+ αv 0. So, u+ αv u+ αv u u + α v u + α v u + α v v 0. Let α, v v the v u v u v u u u v v u u 0 v v + v v v v. Rearragg ths equalty gves u u v v v u wth equalty f ad oly f u vfor soe oplex salar. 9/4/00 Madso Area Tehal College

3 Page 3 La 5 Eleetary Matrx ad Lear Algera Sprg 0 T The adot or Herta ougate of a x oplex atrx s defed as A A,.e., the traspose of the atrx of oplex ougates. Note: whe ovg a atrx ultpler fro left to rght a er produt oe Au v Au v u A v u A v u A v T T T T T trasfors to the Herta ougate. A x oplex atrx U s alled utary f ad oly f U atrx A s alled Herta f ad oly f A A.. U,.e., UU UU I. A x oplex ) Whh of the followg atres are utary? A B C ( ϕ ) ( θ) ( θ) ( ϕ ) ( θ ) ( ϕ ) ( θ) ( θ) ( ϕ ) ( θ ) os( ϕ) 0 s ( ϕ) s os s os os D s s os os s ) What s true aout the deterat of a utary atrx? d) Is every orthogoal atrx a utary atrx? Expla your aswer. e) Look up the defto of a group algera. Show that the set of x utary atres for a group wth respet to the operato of atrx ultplato. f) Let SU() e the set of all x utary atres wth a deterat of oe. Show that SU() s a group wth respet to the operato of atrx ultplato. Fd a eleet of SU() that s ot the detty atrx. 9/4/00 Madso Area Tehal College

4 Page 4 La 5 Eleetary Matrx ad Lear Algera Sprg 0 g) Suppose A s a x atrx ad U s a x utary atrx. What s the det U AU? What s the trae U AU? h) Ca a o-detty atrx e oth Herta ad utary? Expla your aswer. ) For ay atrx elow whh s Herta detere ts egevalues ) Suppose A s x Herta atrx ad u s a vetor. What s Au u Au u? k) Suppose A s x Herta atrx ad for a o-zero ϕ, Aϕ ϕ. What s? Choose oe of the followg sx proles. Eah prole s worth 8 pots. You ay do extra proles to ear up to 40 extra la pots. 4. The Spetral or Fourer Represetato of Herta Matres a) If A s a x Herta atrx the t has learly depedet egevetors whh spa. Let Φ desgate a set of oralzed learly depedet egevetors of A wth AΦ Φ ad { } Φ Φ. Detere Φ Φ. k ) Let B e a x atrx defed as B 'th opoet of Detere BB. Φ,.e., B [ Φ Φ Φ ]. 9/4/00 Madso Area Tehal College

5 Page 5 La 5 Eleetary Matrx ad Lear Algera Sprg 0 Wolfra Alpha a e used to detere the egevalues ad a set of egevetors for a atrx as llustrated elow. ) Detere the egevalues ad a set of orthooral egevetors of the atrx A. 5 3 A 3+ Φ Φ Let v. Copute the followg: v Φ Φ Av 9/4/00 Madso Area Tehal College

6 Page 6 La 5 Eleetary Matrx ad Lear Algera Sprg 0 d) Detere the egevalues ad a set of orthooral egevetors of the atrx A A Φ Φ Φ 3 3 Let v. Copute the followg: 3 v Φ Φ Av e) Let A e a x Herta atrx wth a orthooral set of egevetors { } Φ Φ δ. Let v e ay vetor k k. What does the su v Φ Φ represet? If A s o-sgular, what does the su v Φ Φ represet? Φ so that AΦ Φ ad 9/4/00 Madso Area Tehal College

7 Page 7 La 5 Eleetary Matrx ad Lear Algera Sprg 0 Proles 5 ad 6 are dedated to the eory of Professor C. H. Blahard (93 009) of the Uversty of Wsos-Madso Physs Departet. He was the est ad ost sprg teaher I have ever eoutered. 5. The Moet of Ierta Tesor I the aalyss of the rotato of a rgd ody oe defes a seod rak tesor, alled the oet of erta tesor whh a e represeted y the followg 3 x 3 real syetr atrx. For a ass desty ρ ( x, yz, ) I I I xx xy xz erta xy yy yz I I I I I I I xz yz zz, the atrx eleets of the oet of erta tesor are defed y the followg trple tegrals over the rego of three desoal spae S ouded y the surfae of the rgd ody. xx xy xz yy yz zz S S S S S S (,, ) I ρ xyz y + z dv I I (,, ) (,, ) (,, ) x y z xydv x y z xzdv I ρ x y z x + z dv I ρ ρ ρ (,, ) x y z yzdv (,, ) I ρ xyz x + y dv ωx For a agular veloty vetor ω ω y, the agular oetu of the rgd ody s gve y ω z Iertaω ad the ket eergy of rotato s ω T Iertaω. For a ufor sold rght rular ylder of radus a ad heght h led as show Fgure, the oet of erta tesor s I erta h 5a 3 h a h a h a h 7a /4/00 Madso Area Tehal College

8 Page 8 La 5 Eleetary Matrx ad Lear Algera Sprg 0 Fgure a) Detere the egevalues ad a orthooral set of egevetors for ths I erta. ) The orthooral egevetors are alled the prpal axes of the rgd ody. Expla ths terology. 9/4/00 Madso Area Tehal College

9 Page 9 La 5 Eleetary Matrx ad Lear Algera Sprg 0 6. Coupled Pedulus A sple pedulu ossts of a oet of ass suspeded y a essetally weghtless ad o strethale strg of legth that s allowed to swg freely. As fgure let the agleψ deote the dsplaeet fro the 3π dowwards vertal. ψ s related to the stadard polar agleθ yθ + ψ, so that the posto vetor of the ass 3 π ˆ R 3 π ˆ ˆ ˆ. The aelerato vetor s thus s gve y os + ψ + s + ψ s ( ψ) os( ψ) d R ˆ ˆ dψ s os os ˆ s ˆ d a ψ ψ ψ ψ dt dt dt dψ d ψ s os ˆ dψ d ψ ψ ψ os( ψ) s ( ψ) ˆ dt dt dt dt Fgure ψ 9/4/00 Madso Area Tehal College

10 Page 0 La 5 Eleetary Matrx ad Lear Algera Sprg 0 The fores atg o the ass are the teso the strg, T, whh pots to the org ad gravty whh pots straght dow. F T s( ψ) ˆ+ os( ψ) ˆ g ˆ Ts ( ψ) ˆ+ Tos( ψ) g ˆ. Fro Newto s seod law F a d ψ d ψ d ψ d ψ ˆ dt dt dt dt of oto, s ( ψ) + os( ψ) ˆ+ os( ψ) + s ( ψ) the oupled dfferetal equatos, so that we have dψ d ψ Ts ( ψ) s ( ψ) + os( ψ) dt dt dψ d ψ Tos( ψ) g os( ψ) + s ( ψ) dt dt dψ os ψ d ψ Fro the frst equato T dt. Susttutg ths expresso to the seod equato s ( ψ ) dt gves the followg. ( ψ ) ( ψ ) dψ os d ψ dψ d ψ os g os s dt ψ ψ s dt + ψ dt dt os ( ψ ) d ψ d ψ g s ( ψ ) s ( ψ ) dt dt d ψ d ψ s ( ψ) os ( ψ) g s ( ψ) dt dt d ψ g s( ψ ) dt We seek solutos that satsfy the tal values of ψ ( 0) ad ψ ( 0). Ths o lear seod order tal value prole has a soluto whh a e expressed as a ellpt futo. A spler result s otaed f we restrt our atteto to sall agles where s ψ ψ. The dfferetal equato the s lear ad s gve y d ψ g ψ. dt 9/4/00 Madso Area Tehal College

11 Page La 5 Eleetary Matrx ad Lear Algera Sprg 0 t Lookg for expoetal solutos of the for ψ () t e g gves the auxlary equato whh has solutos g ± ± ω0 ω0t ω0t. Thus, ψ () t e e ( ) os( ω t) ( ) s ( ω t) soluto e real eas that ψ () t ψ () t Requrg that the 0 0, so that ( ) os( ω t) ( ) s ( ω t) ( ) os( ω t) ( ) s( ω t) Se se ad ose are learly depedet futos t ust e true that + +. Takg, d + d for real,, d, d yelds d ad d so that + ad. So the geeral soluto the sall agle approxato s ψ () t os( ω t) s ( ω t) the tal odtos gves () ( 0os ) ( 0) 0 0 ω0 +. Fttg 0 0 ψ ψ t ψ ω t + sω t. The soluto s perod wth a perod gve y π π g T ad a frequey of f ω g T π 0 () t ( 0os ) ( t), so that ψ ( 0) ψ ψ ω 0. If the pedulu s released fro rest, ψ ( 0 ) 0ad s the apltude of the agle osllatos. Two pedulus eah of legth are oupled y a ross strg as show o Fgure 3. The effet of the ross strg a e odeled y a fore o pedulu y pedulu equal to k ( ψ ψ ) where k s a postve proportoalty ostat. By Newto s thrd law of oto the fore o pedulu y pedulu s k ( ψ ψ ) agle approxato the two agles satsfy the followg lear dfferetal equato. g k k d ψ ψ dt ψ g ψ k k. I the sall 9/4/00 Madso Area Tehal College

12 Page La 5 Eleetary Matrx ad Lear Algera Sprg 0 Fgure 3 Graph fro page 7 of Ne Experets for a Thrd Grade Hour, 009 Meoral Edto y C. H. Blahard. Fgure 4 9/4/00 Madso Area Tehal College

13 Page 3 La 5 Eleetary Matrx ad Lear Algera Sprg 0 g k k Let A. A s a real syetr atrx. It has orthooral egevetors Φ, Φ that spa g k k R. So for ay value of t, () t () t ψ () t Φ + () t Φ ψ. Thus, ψ d d Φ + Φ A Φ + Φ ψ d dt dt dt. a) Dsplay the seod order dfferetal equatos for oth () t ad () t. ) Solve for oth () t ad () ad ψ ( 0 ). 0, 0, 0, t wth tal odtos for eah defed y the values of ψ ψ ψ ) Use the solutos for () t ad () t to dsplay the solutos for ψ () t ad ψ () t. d) Is the oto of the oupled pedulus always perod,.e. s there a perod T wth oth ψ ( t T) ψ ( t) ψ ( t T) ψ ( t) + ad + for all values of t? Expla your aswer. What odtos o the ostats k,, gs requred for perodty? 9/4/00 Madso Area Tehal College

14 Page 4 La 5 Eleetary Matrx ad Lear Algera Sprg 0 e) Dsplay the solutos for ψ () ad () t ψ f oth asses are released fro rest,.e. ψ ψ t f) If ψ ( 0) ψ ( 0) 0ad ψ 0 0 desre the oto. I partular, does the seod pedulu osllate? g) Dsplay the solutos for ψ () ad () t ψ f ψ ( 0 ) ψ ( 0 ) 0ad ( 0) ( 0) t ψ ψ ψ. 0 h) Dsplay the solutos for ψ () ad () t ψ f ψ ( 0 ) ψ ( 0 ) 0ad ( 0) ( 0) t ψ ψ ψ. 0 ) The solutos of g) ad h) are perod solutos whh osst of osllatos at a sgle frequey. These are alled the oral odes of the syste. Oe of these odes s alled a walkg ode. Whh oe s t? ) Oe of the oral odes s faster,.e. has a hgher frequey. Whh oral ode s t? k) What s the oeto etwee the egevetors of A ad the oral odes? l) What s the oeto etwee the egevalues of A ad the oral odes? 9/4/00 Madso Area Tehal College

15 Page 5 La 5 Eleetary Matrx ad Lear Algera Sprg 0 7. Raylegh-Rtz Varatoal Calulatos of Extree Egevalues Suppose A s a x real syetr atrx. I ay applatos suh as oud states Quatu Mehas ad eoo optzatos the extree egevalues ad are of partular terest. Cosder a x vetor Ψ alled a tral futo. The egevetors of A are learly depedet ad a orthooral ass for R osstg of egevetors of A s always possle. Desgate ths orthooral ass y { } AΦ Φ, ( Φ, Φ k) δ k, ad (, ). Ψ ΨΦ Φ Φ wth. The tral futo a e expaded as ( AΨΨ, ) A ( ΨΦ, ) Φ, ( ΨΦ, k) Φ k ( ΨΦ, )( ΨΦ, k)( AΦ, Φk) k k (, )(, k)(, k) (, )(, k) δ k (, ) ΨΦ ΨΦ Φ Φ ΨΦ ΨΦ ΨΦ k k ( A, ) Ψ, the Raylegh Quotet s defed as ( ΨΨ) For 0 ΨΨ,, ( ΨΦ, ) ( ΨΨ). It a e raketed as follows: Ay hoe of Ψ 0 zero vetors of ( ΨΦ ) ( ΨΦ ) ( AΨΨ, ) ( AΨΨ, ) ( AΨΨ, ) ( ΨΨ, ) ( ΨΦ ),, ΨΨ, ΨΨ, ΨΨ, ( ΨΦ ),, ΨΨ, ΨΨ, ΨΨ, gves a lower oud to ad a upper oud to. By allowg Ψ to vary over all o R we have the followg haraterzatos. Ψ R, Ψ 0 Ψ R, Ψ 0 ( AΨΨ, ) ( ΨΨ, ) ( AΨΨ, ) ( ΨΨ, ) 9/4/00 Madso Area Tehal College

16 Page 6 La 5 Eleetary Matrx ad Lear Algera Sprg 0 If Ψ vares over ust a suspae W of R the Ψ W, Ψ 0 ( AΨΨ, ) ( ΨΨ, ) ad AΨΨ, Ψ W, Ψ 0 (, ) ΨΨ. If Φ W the Ψ W, Ψ 0 ( AΨΨ, ) (, ) ΨΨ or f Φ W the Ψ W, Ψ 0 ( AΨΨ, ) ( ΨΨ, ). How well the varatoal alulatos Ψ W, Ψ 0 ( AΨΨ, ) ( ΨΨ, ) ad AΨΨ, Ψ W, Ψ 0 ( ΨΨ, ) approxate the extree egevalues depeds o how lose the suspae W s to Φ ad Φ.Let Η Ψ ( ΨΨ, ), the A, ( ΨΨ, ) ΨΨ ( A Η, Η ) wth ( ΗΗ, ). So prate varatoal alulatos are doe usg oralzed tral futos,.e. ( ΨΨ, ). Ths s equvalet to varyg the tral futo over a suspae of the ut -sphere. As a exaple osder the x real syetr atrx a A. The egevalues are solutos of the haraterst polyoal 4 a+ + a + a+ a a+ a + a+ a + a+ + a 0. os θ The tral futo Ψ s ( θ ) wth 0 θ < π spas the ut rle (the ut -sphere). Thus, ( A Ψ, Ψ ) ad ( A, ) 0 θ< π Ψ Ψ. Ths a e verfed as follows. 0 θ< π + + aos θ s θ os θ f ( AΨ Ψ ) a + + os θ s θ s θ θ,, os θ os θ s θ s θ Usg the trgooetr dettes os ; s ; s os 9/4/00 Madso Area Tehal College + os θ os θ s θ θ ( θ) ( θ) ( θ)

17 Page 7 La 5 Eleetary Matrx ad Lear Algera Sprg 0 f ( a ) a+ a θ + os( θ) + s ( θ) ; os ( δ) ; s ( δ) a a + + a+ a a+ a + + os ( δ) ( θ) ( δ) ( θ) os os s s ( θ δ) So the u of f ( θ ) at δ θ s + + a+ a ad the u of f ( θ ) at δ π θ + s 4 + a+ a. For a 3 x 3 real syetr atrx, tral futos of the for Ψ os s 0 ( θ ) ( θ ) would geeral gve oly approxatos to ad se Ψ does ot spa all of the ut 3-sphere. The tral futo Ψ s s ( ϕ ) os( θ ) ( ϕ ) s ( θ ) os( ϕ ) wth 0 θ < π ;0 ϕ π does spa the ut 3-sphere ad so ( AΨ, Ψ ) ad ( A, ) 0 θ< π 0 ϕ π Ψ Ψ. 0 θ< π 0 ϕ π For a 4 x 4 real syetr atrx, tral futos of the for ( α ) ( ϕ ) ( θ ) ( α ) ( ϕ ) ( θ ) s ( α) os( ϕ) os( α ) s s os s s s Ψ wth 0 θ < π ;0 ϕ π ;0 α π would gve the exat values of extree values of a futo of three depedet varales. ad, ut ths would volve fdg 9/4/00 Madso Area Tehal College

18 Page 8 La 5 Eleetary Matrx ad Lear Algera Sprg 0 Fll the followg tales for eah atrx elow. a) 0 6 A 6 5 os Ψ s ( θ ) ( θ ) f ( θ ) ( AΨ, Ψ ) f f ) 7 4 A os Ψ s ( θ ) ( θ ) f ( θ ) ( AΨ, Ψ ) f f ) 5 3 A 3+ os Ψ s ( θ ) ( θ ) f ( θ ) ( AΨ, Ψ ) f f 9/4/00 Madso Area Tehal College

19 Page 9 La 5 Eleetary Matrx ad Lear Algera Sprg 0 d) A ( θ ) ( θ ) os Ψ s 0 f ( θ ) ( AΨ, Ψ ) f f e) A ( ϕ ) os( θ ) ( ϕ ) s ( θ ) os( ϕ ) s Ψ s f ( ϕθ, ) ( AΨ, Ψ ) f f f) A ( θ ) os 0 Ψ s ( θ ) 0 f ( θ ) ( AΨ, Ψ ) f f 9/4/00 Madso Area Tehal College

20 Page 0 La 5 Eleetary Matrx ad Lear Algera Sprg 0 g) A ( ϕ ) os( θ ) s 0 Ψ s ( ϕ ) s ( θ ) os( ϕ ) f ( ϕθ, ) ( AΨ, Ψ ) f f h) For soe of the aove ases the terval [, ] was ot detal to[ f, f ]. Expla why eah ase. ) ( pot ous) The Raylegh-Rtz varatoal ethod a e used to fd the extree egevalues of oplex Herta atres. Expla how the tral futos eed to e hose to ake ths work, 8. Iteratve Shee to Calulate Egevalue of Greatest Asolute Value Cosder a real x atrx A wth dstt real egevalues. Let Λ e the ost extree egevalue (.e., the egevalue of largest asolute value) of A wth egevetor Φ ad desgate the reag egevetors of A as{ Φ k} k wth AΦ k kφ k. Let Ψ e a vetor R. Beause the egevetors of A for a ass for R, Ψ has a expaso the egespae of A: Ψ Φ + kφk. Assue that k 0. The ( A ΨΨ, ) A kφk, Φ Λ ( Φ, Φ ) + k k( Φk, Φ ). 9/4/00 Madso Area Tehal College k k k

21 Page La 5 Eleetary Matrx ad Lear Algera Sprg 0 k Se < Λ + ( A, ) ( A ΨΨ, ) (, ) (, ) Λ Φ Φ + Φ Φ ΨΨ Λ Φ Φ + Φ Φ + + k k k k (, ) k k ( k, ) k + + k Λ k Λ k (, ) k ( k, ) Λ Φ Φ + Φ Φ k (, ) k ( k, ) Λ Φ Φ + Φ Φ + k ( Φ, Φ ) + k ( Φk, Φ ) k Λ Λ k ( Φ, Φ ) + k ( Φk, Φ ) k Λ Λ, all of the sus k k ( Φk, Φ ) ad k (, ) k Φk Φ wll approah zero k ( A + ΨΨ, ) as reases. Thus as reases the rato ( A ΨΨ, ) Λ + k overges to Λ. Fll the followg tales. a) 0 6 A 6 5 Λ ( A + ΨΨ, ) ( A ΨΨ, ) Ψ /4/00 Madso Area Tehal College

22 Page La 5 Eleetary Matrx ad Lear Algera Sprg 0 ) 7 4 A Λ ( A + ΨΨ, ) ( A ΨΨ, ) Ψ ) 7 4 A Λ ( A + ΨΨ, ) ( A ΨΨ, ) Ψ /4/00 Madso Area Tehal College

23 Page 3 La 5 Eleetary Matrx ad Lear Algera Sprg 0 d) A Λ ( A + ΨΨ, ) ( A ΨΨ, ) Ψ e) A Λ ( A + ΨΨ, ) ( A ΨΨ, ) Ψ f) Why are the results for ) ad ) dfferet? 9/4/00 Madso Area Tehal College

24 Page 4 La 5 Eleetary Matrx ad Lear Algera Sprg 0 9. Egevalues of a Trdagoal Toepltz Matrx Cosder the y atrx G(), defed y the followg forula: G aδ + δ + δ,, +,,,,.e., G a a a a a a a Matres suh as G() whose values alog a gve dagoal are ostat are referred to as Toepltz atres. If 0 the oly egevalue of G() s a. To get a ore useful result we ust assue that 0. Let X e a opoet egevetor of G() wth egevalue. The opoets of X satsfy the equato ( a ) X k+ + X k + X k k 0 () wth the oudary odtos that 0 X + X 0. () Equato () s a lear hoogeeous reurree relato of degree ad t a e solved a aer aalogous wth solvg a lear seod order dfferetal equato. Se X 0 0 ad a X k + X k + X k, a otrval (.e., ozero) egevetor requres that X e ozero. Wthout loss of geeralty we a take X. For a gve egevalue ths tal odto uquely deteres X. To see ths, suppose Y s a seod egevetor wth Y. Beause equato () s lear ad hoogeeous, Z Y X also satsfes equato (). However, Z 0 Z 0 so that the reurso requres that Z s the zero vetor. 9/4/00 Madso Area Tehal College

25 Page 5 La 5 Eleetary Matrx ad Lear Algera Sprg 0 To ostrut the geeral soluto of equato () we look for expoetal solutos of the for k r X k wth r 0. r k + + whh gves the two solutos Hee, r ( a ) r 0 a r + a ad r a a. If the dsrat s zero, r r ad the geeral soluto of equato () would e of the for k k X k α r + β kr for ostats α ad β. However, the oudary odtos of equato () would fore oth ostats to e zero. Thus, a otrval egevetor requres that r r. For ths ase the geeral soluto of equato (4) s gve y k k X k α r + β r. The tal odtos X 0 0 ad X are satsfed whe α β. Usg the equatos for r ad r to solve for gves r r a + + r a + + r. (3) r r Ths results the followg quadrat equato for r, r r + r + 0, so that r + r ± r r r r. Se r r, t ust e true that r r. The oudary odto at X r r, so that for ay teger, k + a e expressed as ( r r ) 0 + r + π r e. Ths equato has the followg dstt solutos eah of whh orrespods to a dfferet otrval egevetor: π r r e + π π π for. Hee, + r r r e e +, so that r e +, where the radal desgates the prpal square root of a oplex uer. Fro equato (3) the egevalues of G() are gve y 9/4/00 Madso Area Tehal College

26 Page 6 La 5 Eleetary Matrx ad Lear Algera Sprg 0 9/4/00 Madso Area Tehal College os a e e a π π π. (4) It eeds to e oted that for oplex ad t s ot eessarly true that. The result ould e off y a us sg (reall that ). O the other had, π π π π a a a os os os, so that allowg the dex to ru fro to the expresso a + + os π wll deed geerate all the egevalues of G(). Copute ad dsplay the fve egevalues ad assoated egevetors of the followg atrx