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Note. Please review 組合矩陣理論筆記 (4) 1-3 about. Jordan form and Minimal polynomial. Theorem 1.2.13. If m=n ,and at least one of A or B is nonsigular,then. AB and BA are similar. Remark. The Jordan structure of AB and BA . corresponding to nonzero eigenvalues are. the same . Qestion p.1.

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Note

Please review 組合矩陣理論筆記(4) 1-3 about

Jordan form and Minimal polynomial


Theorem 1 2 13
Theorem 1.2.13

If m=n ,and at least one of A or B is nonsigular,then

AB and BA are similar


Remark
Remark

The Jordan structure of AB and BA

corresponding to nonzero eigenvalues are

the same


Qestion p 1
Qestion p.1

Given

when does there exist matrices

such that

and


Qestion p 2
Qestion p.2

Solved in

H.Flanders, Elementary divisors of AB and

BA Pra. Amer. Math. Sec 2(1951) 871-874

C.R Johnson, E.A.Schreineer, The

relationship between AB and BA, Ameri

Math Monthly 103(1966),578-582


Solution p 1
Solution p.1

A,B exist if and only if

(i) The Jordan structure associated with

nonzero eigenvalues is identical in C and

D


Solution p 2
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)


Problem
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


Equivalent problem
Equivalent Problem

Given

when does

there exist a matrix A in the superdiagonal

--block form, i.e.

such that


Theorem p 1
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


Theorem p 2
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



Cyclically consecutive equal components
Cyclically consecutive equal components

is said to have cyclically

consecutive equal components, if whenever

with

then

either

or


Example 1
Example 1

has cyclically consecutive equal

components


Example 2
Example 2

has no cyclically consecutive equal

components


Theorem p 11
Theorem p.1

Given square matrices

there are

rectangular matrices

such that

iff (a)

have the same

subcollection of nonsigular elementary

Jordan blocks.


Theorem p 21
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


Theorem p 3
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.



Example 1 p 1
Example 1 p.1

Let

A List of nilpotant elementary Jordan blocks

the corresponding

2-tuple :


Example 1 p 2
Example 1 p.2

Let

A List of nipotant elementary Jordan blocks

the corresponding

2-tuple :


Example 2 p 1
Example 2 p.1

Given

A list of nilpotent elementary Jordan blocks

satisfy the condition (b) in Theorem

(see next page)


Example 2 p 11
Example 2 p.1

The corresponding 4-tuple are


Jordan diagram
Jordan Diagram

Let T be a nilpotent operator on V and 5 is

the index of nilpotency of T.

Jordan chain for T


Jordan diagram1
Jordan Diagram

a basis for N(T)


Jordan diagram2
Jordan Diagram

a basis for N(T2)


Jordan diagram3
Jordan Diagram

a basis for N(T3)


Jordan diagram4
Jordan Diagram

a basis for N(T4)=V



Theorem
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


Observation 1 p 1
Observation 1 p.1

Let C,D be m-cyclic matrices (not necessarily

of the same size) in superdiagonal block form

say,

and if



Observation 2 p 1
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


Observation 2
Observation 2

Then there exist an rxr permutation matrix P

such that

is an m-cyclic matrix

block form

in the superdiagonal

and


Spectral i
Spectral I

Let

and

The spectral index of A =

: = g.c.d of i at



ad