Page 114 - Textos de Matemática Vol. 39

104 THOMAS LAFFEY AND HELENA SˇMIGOC
Then
cA(x) − g(x) = −a21a3 + b21 + 2a1c1 + (−72 + 2a1b1 + 8t2)x
+(a21 + 2a1a3 − 2(30 + c1) + 12t2)x2 + (10 − 2b1 + 6t2)x3
+(15−2a1 −a3 +t2)x4.
We see that if we want the matrix A to have the characteristic polynomial g(x) the following equations must hold:
b1 = a1 =
a3 = c1 =
5+3t2, 4(9 − t2)
5+3t2, 3+58t2 +3t4
5+3t2 ,
2(3 + 591t2 − 103t4 + 21t6)
(5+3t2)2 ,
−331 + 20127t2 − 270t4 + 11358t6 + 1641t8 + 243t10
0=
The last equation is satisfied for
t3 = 0.128245...
Since a1, a3, b1, c1 are nonnegative for t3 we conclude that σ3 can be realized for t = t3.
The smallest t for which σ2 = (3+t, 3−t, −2, −2, −2) is the spectrum of a symmetric nonnegative matrix is t = 1. Now we consider the question for which t is σ3 = (3 + t, 3 − t, −2, −2, −2, 0) the spectrum of a symmetric nonnegative matrix. Matrix
⎡2 0 22 0⎤
⎢ 3 ⎢02022⎥
⎥ ⎢ 3⎥
⎢22 0 0 4⎥ ⎣33⎦
02240 33
has spectrum (10, 8,0,−2). We use Theorem 4.1 to join lists (10, 8,0,−2) and 33 33
(2, −2). Now we know that the list ( 10 , 8 , 0, −2, −2) is realizable by a nonnega- 33
tive symmetric matrix with a diagonal element 2. Joining lists ( 10 , 8 , 0, −2, −2) 33
and (2,−2) proves that σ3 is realizable with t4 = 1. We do not know if t4 = 1 33
is the smallest possible t that will make σ3 realizable by a symmetric matrix. References
[1] Louis Block, John Guckenheimer, Michal Misiurewicz, and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical