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6th - EIEHGISE SEI 4.4 is{a ﬁh}={h l{A—5...

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Unformatted text preview: EIEHGISE SEI' 4.4 is. {a} Theeharaeterhtioptdynomialiaﬁh}={h+l}{A—5}. Thccigcnuclncccrca- —1 ma=a {b} Thecharacterhﬂepolynomialispﬂi = (Ji- 3H). — '1'}th — l}. Theeigenvalneanre A - 3. A = T, and .1: 1. (c) The characteristic polynomial is 13(1) = (Ji+ iizﬂ - ”(A — é}. The eigenvalues an A = “1} {with multiplicity 2]. J. = I. and .t = g. —E 1. l o 1 o 53 —5 il . .- 1 ,cnd [T2] mpe-ctweh'. Thc-gcccclccccfai arch: (1)9: 1, 14%)”: 3+}, A: to)” :41 ﬂ 1 Elﬁn—- II E! 18. TheagmvaloeaoanreA=lJi=%.A-ﬂ,ahdh=2.ﬂorreepoodjngeigenvectoraaie[ and J. = [2F = 512. Corresponding eigenvectora are the same as abmre. as. The characteristic pohmom'lal of A is pH] = u. — 330.3 — 2A: + c9 _ 4}. Note that ties ccccnd actor in this polynomial cannot have a double root {for an},r value of it} since [-2:c-:I2 — 4(a- — 4) =- 16 ;£ ii. "I'hus the onl}r While repeated eigenvalue ofA isJi=3, andthis occursii‘aasiuiouil}r lifh=iiiara rootoi'theseceoﬂ factor ofﬁh], i.e. ifandonlyifQ-ﬁz+:r’—4=ﬂ. The mint-ﬂ?“ quadratic equation area: laods=5. Fbrthese valuesofmJ-ﬂisaneigenvaloeoi'moltiphcityﬂ. 111. no. {a} Fate-Farmmrle.thereduwdmmm‘°m°”'li 21"”:[1]: 3] {'1} True. We have {“11 + In] I 3111 + \$23!: and. if .11 75 in il- EBJI be 311MB (since 11 and In must be lines-ﬂy hidependent} that Jun; + Aux: gé Mac; + x2] for any value of 13. [c] Tine. The characteristic polars-main] of A is a cubic polynomial, and every cubic polynomial has at least one real MM. {Ii} Thee. Hpﬂ} = 1” + 1.1:th detIA} = {—rl}“p[ﬂ} - :|:l 7‘3 {1; thus A is invertible. P3. Suppose that Ax = Jo: where 1' 5E El and A is invertible. Then 3: .= ill—12b: = iii-Us: =- Ari—1:: and,aineeh;éﬂ[becauaeidiainmtihle). it followathat A“‘x=§x. Thuaiiaaneigsmralue of A" and x is a corresponding eigenvector. CHAPTER 5 Matrix Models EXERCISE SET 5.3 5. {a} In the secoml now. we haw |-‘2| = 2. This matrix is not strictlyr clingmiallar dominant. {in} and {c} are strictly diagonally dominant. CHAPTER s Linear Transformations EXERCISE SET E1 12. [a] and {c} are linear transformations. [b] is not linear; neither homogenemn nor additive. wet -: -1] .( '3 )= 3'3-‘1l'ii'31 wharaooaﬂ- mudﬂiﬂE': FIT—I11!- ﬂjnﬂ will ain’ﬂ lﬂnFMF 31+ {“3 H15: 3min M-mm] [:aln;mﬂ Ali-12'9- m7ﬂ' _ 2 WHL=ﬁF=m m “21:11 maﬁa mama J I 1 m {'1} PL 2 [ainﬂ‘maﬂ shill - ”"15 [In 111?] D3. Fran:- Eamiliar tﬁganamatrir. Hamming, we have A - [2: _:: = R”. Thus multiplication byﬁmrmpandatnmtaﬁon about theariginthroughthaangleiﬂ. D5. Siaaa Tm} = 3m 5'5 I], th'u tranaﬁarmation '15 mt linear. Gmmelrically, it myomhta a rotation followed by a translation. ...
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6th - EIEHGISE SEI 4.4 is{a ﬁh}={h l{A—5...

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