1. Note Please review 組合矩陣理論筆記(4) 1-3 about
Jordan form and Minimal polynomial

2. Theorem 1.2.13 If m=n ,and at least one of A or B is nonsigular,then
AB and BA are similar

3. Remark The Jordan structure of AB and BA
corresponding to nonzero eigenvalues are
the same

6. Qestion p.1
Given
when does there exist matrices
such that
and

7. Qestion p.2 Solved in H.Flanders, Elementary divisors of AB and
BA Pra. Amer. Math. Sec 2(1951)
C.R Johnson, E.A.Schreineer, The
relationship between AB and BA, Ameri
Math Monthly 103(1966),

8. Solution p.1 A,B exist if and only if
(i) The Jordan structure associated with
nonzero eigenvalues is identical in C and D

9. Solution p.2 (ii) If are the sizes of the
Jordan blocks associated with 0 in C while
are the corresponding
for all i
sizes in D, then
( Here, for convenience, we fill out lists
of zero Jordan blocks sizes with 0 as
needed)

10. Problem Given square complex matrices
not necessarily of the same size. Find a
necessary and sufficientary condition on
so that there exist complex
rectangular matrices
of appropriate size that satisfy

11. Equivalent Problem Given when does
there exist a matrix A in the superdiagonal
--block form, i.e.
such that

12. Theorem p.1 Let U(A):= the collection of elementary Jardan
blocks in the Jordan form of A . Given
To obtain
from U(A)
replace each by
and a by
k times if

13. Theorem p.2 and by m-q copies of together with q copies of
where p is a positive integer and q is a
nonnegative integer determined uniquely by

14. Example

19. Cyclically consecutive equal components
is said to have cyclically
consecutive equal components, if whenever
with
then
either or

20. Example 1
has cyclically consecutive equal
components

21. Example 2
has no cyclically consecutive equal
components

22. Theorem p.1 Given square matrices there are rectangular matrices
such that
iff (a)
have the same
subcollection of nonsigular elementary
Jordan blocks.

23. Theorem p.2 (b) It is possible to list the nilpotent
elementary Jordan blocks in
in some way, say,
are nonnegative
where
(and
stands for an empty block)
so that for each positive integer

24. Theorem p.3 is either an m-tuple
with constant components, or an m-tuple
with two distinct components that differ by 1
and in which equal components are cyclically
consecutive.

35. Explain for (b)

36. Example 1 p.1 Let A List of nilpotant elementary Jordan blocks
the corresponding
2-tuple :

37. Example 1 p.2 Let A List of nipotant elementary Jordan blocks
the corresponding
2-tuple :

38. Example 2 p.1 Given A list of nilpotent elementary Jordan blocks
satisfy the condition (b) in Theorem
(see next page)

39. Example 2 p.1
The corresponding 4-tuple are

40. Jordan Diagram Let T be a nilpotent operator on V and 5 is
the index of nilpotency of T.
Jordan chain for T

41. Jordan Diagram
a basis for N(T)

42. Jordan Diagram
a basis for N(T2)

43. Jordan Diagram
a basis for N(T3)

44. Jordan Diagram
a basis for N(T4)=V

45. Jordan Diagram

46. Theorem Let V be a finite dimensional vector space Let
be a nilpotent operator
then there is an ordered basis β of V s.t.
is a Jordan matrix

51. Observation 1 p.1 Let C,D be m-cyclic matrices (not necessarily
of the same size) in superdiagonal block form
say,
and if

52. Observation 1 p.2
then

53. Observation 2 p.1 Let be a given m-tuple
of positive integers whose components take
on at most two distinct values that differ by 1
and in which equal components are cyclic
consecutive. Denote
Then

54. Observation 2 Then there exist an rxr permutation matrix P such that
is an m-cyclic matrix
block form
in the superdiagonal
and

56. Spectral I
Let
and
The spectral index of A =
: = g.c.d of i at

57. Combinatorial Spectral Theory of Nonnegative Matrices