7.3hwpage1 - 17,3 14 24 28 FURMfi E{W U 1 U...

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Unformatted text preview: 17,3 14. 24. 28. FURMfi. E {W U 1 U A1:0,A1:A3:1,eigenba5is:r 1 , —5 , 2 0 a b 0 a 0 20. F01" A1 = 1, E1 = her 0 0 c = ker 0 fl .1 so ifa = (J then E1 is 2-dimeneional, 0 O 1 0 0 0 otherwise it is 1-dimensional. 1 —-a. —b For A2 : 2, E3 = ker 0 1. -—r: so E2 is 1-dimensional. O 0 0 Hence, there is an eigeubasis if a = 0. LetA: 0‘ b.Firstwewant “ b 2 : 2 ,or2a+b=2,‘2c+d=1. This c of c d 1 1 condition is satisfied by all matrices of the form A = 2' i : g: . Next, we want there to be no other eigenvalue, besides 1I so that 1 must have an algebraic multiplicity of 2. We want the characteristic polynomial to be {A — 1)2 = A2 -— 2A + 1, so that the trace must be 2, and u + (1 i 20) = 21 or, a = l + 2c. Thus we want a. matrix of the form _ 1 + 2c —4c A # l c 1 - 2c ' Finally, we have to make sure the E; = span 3 instead of E1 = R2. This means that we must exclude the case A = I2. In order to ensure this, we state simply that A: [l—i—Zc —4c 0 1 2c] , where c is any nunzem cum-taut Since Jaw) is triangular, its eigenvalues are its diagonal entries, hence its only eigenvalue is 36:. Moreover, 0 I U 0 U 0 1 ' E]; = ker(.]fl (k) — kIn) = leer ' Q = span (El). : : f 1 0 0 U 0 The geometric multiplicity of k: is 1 while its algebraic multiplicity is n. ...
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