By Badiale M., Benci V., Rolando S.

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The 4-step paths that begin and end at node 2 are 2 to 1 to 1 to 1 to 2, 2 to 1 to 2 to 1 to 2, and 2 to 1 to 3 to 1 to 2. Harder to find the eleven 4-step paths that start and end at node 1. Notice the column times row multiplication above. Since A = AT the eigenvectors in the columns of S are orthogonal. They are in the rows of S −1 divided by their length squared. 40 B has the same  eigenvectors   (1, 0) and (0, 1)as A, so B is also diagonal. The 4 equations a b a 2b 0 0 − =  have coefficient matrix with rank 2.

14 The rule Fk+2 = Fk+1 + Fk produces the pattern: even, odd, odd, even, odd, odd, . . 15 (a) True (b) False (c) False (might have 2 or 3 independent eigenvectors). 16 (a) False: don’t know λ (b) True: missing an eigenvector (c) True.       8 3 9 4 10 5  (or other), A =  , A =  ; only eigenvectors are (c, −c). 17 A =  −3 2 −4 1 −5 0 18 The rank of A − 3I is one. Changing any entry except a12 = 1 makes A diagonalizable. 19 SΛk S −1 approaches zero if and only if every |λ| < 1; B k → 0.

7 The columns of S are nonzero multiples of (2, 1) and (0, 1) in either order. Same for A−1 .   a b  for any a and b. 8  b a       2 1 3 2 5 3 2 3 4 , A =  , A =  ; F20 = 6765. 5) −3 3 3 3          2 1 2 Gk+1 G 3 3 1 3  = Ak  1  →  (c)      =  . 2 1 2 Gk G0 0 3 3 3 52      λ λ λ 0 1 −λ 1 2 1 2 1 =    . 11 A = SΛS −1 =  λ1 − λ2 1 1 0 1 0 λ2 −1 λ1        k λ λ λ 0 1 −λ 1 − 1 2 2 1   1    =  . SΛk S −1 = λ1 − λ2 1 1 0 λk2 −1 λ1 0 (λk1 − λk2 )/(λ1 − λ2 ) 1 1  12 The equation for the λ’s is λ2 −λ−1 = 0 or λ2 = λ+1.

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A nonlinear elliptic equation with singular potential and applications to nonlinear field equations by Badiale M., Benci V., Rolando S.


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