Page 130 - Matrices theory and applications

6.4. Exercises
With the notation of the previous exercise, show that the set of
solutions of (6.3) is spanned by the solutions of the form
m
(a R(m)) m∈IN ,
9. Let n ≥ 2and let M ∈ M n (ZZ) be the matrix defined by m ij =
i + j − 1:
 R ∈ CC[X], deg R< n a . 113

1 2 ··· n
 . . . . . . 
 2 . . . 
M =  . .  .

 . . . . . . . . . . 

n ··· ··· 2n − 1
(a) Show that M has rank 2 (you may look for two vectors x, y ∈ ZZ n
such that m ij = x i x j − y i y j ).
(b) Compute the invariant factors of M in M n (ZZ) (the equivalent
diagonal form is obtained after five elementary operations).
10. The groundfieldis CC.
(a) Define
 
... 0 1
 .
 . . . . . . . . 0 

N = J(0; n), B =   .
 . . . . . 
 0 . . . . 
1 0 ...
1
Compute NB, BN,and BNB. Show that S := √ (I + iB)is
2
unitary.
(b) Deduce that N is similar to
0 1 0 ... 0  0 ... 0 −1 0 
 
. . .
 . . . .  .
1 . . . . .   . . . . 
 . . .
 . . . . 1 
 
. . . . . .  i  . . . . . . 
1 
 0 . . . 0  +  0 . . . 0  .
2   2  
 . . . .   . . . . . . . 
 . . . . . . . . 1   −1 . . . . . 
0 1 0 ... 0
0 ... 0 1 0
(c) Deduce that every matrix M ∈ M n (CC) is similar to a complex
symmetric matrix. Compare with the real case.