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 Given when does there exist matrices such that and

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

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

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 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

Example

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

Example 1 has cyclically consecutive equal components

Example 2 has no cyclically consecutive equal components

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.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 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.

Explain for (b)

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

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

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

Example 2 p.1 The corresponding 4-tuple are

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

Jordan Diagram a basis for N(T)

Jordan Diagram a basis for N(T2)

Jordan Diagram a basis for N(T3)

Jordan Diagram a basis for N(T4)=V

Jordan Diagram

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

Observation 1 p.2 then

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

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

Combinatorial Spectral Theory of Nonnegative Matrices