Page 128 - Matrices theory and applications

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6.4. Exercises
r
invariant factors are thus 1,... , 1, (X − a) . Hence we have the following
theorem.
r
Theorem 6.3.7 When an elementary divisor of M is (X − a) ,one may,
in the second canonical form of M, replace its companion matrix by the
Jordan block J(a; r).
Corollary 6.3.1 If the characteristic polynomial of M splits over k,then
M is similar to a block-diagonal matrix whose jth diagonal block is a Jordan
block J(a j ; r j ). This form is unique, up to the order of blocks.
Corollary 6.3.2 If k is algebraically closed (for example if k = CC), then
every square matrix M is similar to a block-diagonal matrix whose jth
diagonal block is a Jordan block J(a j ; r j ). This form is unique, up to the
order of blocks.
6.4 Exercises
See also the exercise 12 in Chapter 7.
1. Show that every principal ideal domain is a unique factorization
domain.
2. Verify that the characteristic polynomial of the companion matrix of
a polynomial P is equal to P.
3. Let k be a field and M ∈ M n (k). Show that M, M T have the same
T
rank and that in general, the rank of M M is less than or equal
to that of M. Show that the equality of these ranks always holds if
k = IR, but that strict inequality is possible, for example with k = CC.
4. Compute the elementary divisors of the matrices
   
22 23 10 −98 0 −21 −56 −96
12 18 16 −38 18 36 52 −8
   
  ,  
−15 −19 −13 58 −12 −17 −16 38
   
6 7 4 −25 3 2 −2 −20
and
 
44 89 120 −32
0 −12 −32 −56
 
 
−14 −20 −16 49
 
8 14 16 −16
in M n (CC). What are their Jordan reductions?
5. (Lagrange’s theorem)