Spectrum of A σ(A)=the spectrum of A =the set of all eigenvalues of A

Proposition

Similarity matrix If,then we say that A is transformed to B under similarity via similarity matrix P

Exercise If are similar over C, then A and B are similar over R. 組合矩陣理論 第一 章 Exercise.doc

Proof of Exercise 1.2.4

Schur’s unitary triangularilation Theorem unitarily similar can be in any prescribed order

Normal matrix e.g Hermitian matrix, real symmetric matrix, unitary matrix, real orthogonal matrix, skew- Hermitian matrix, skew-symmetric matrix. 強調與 complex symmetric matric 作區 別

Remark about normal matrix Normal matices can not form a subspace.

Fact (*) for Normal matrix Proof in next page

Spectrum Thm for normal matix 注意 Appling Schur’s unitary triangulariation Theorem to prove.

Real Version of Spectrum Thm for normal matix It is normal. The proof is in next page

Proposition for eigenvalue

Proof of privious Proposition

1.3 Jordan Form and Minimal Polynomial

Elementary Jordan Block main diagonal elementary jordan block super diagonal sub diagonal

It is Nilpotent matrix.(see next page)

Jordan Matrix jordan matrix

Jordan Canonical Form Theorem unique up to the ordering of elementary Jordan blocks along the block diagonal. A is similar to a jordan matrix If A is real with only real eigenvalues, then the similarity matrix can be taken to be real By Exercise 1.2.4

Observation 1 for Jordan matrix the jordan matrix of A

Observation 2 for Jordan matrix the proof in next page

Observation 3 for Jordan matrix

Observation 4 for Jordan matrix

Observation 5 for Jordan matrix Given counter example in next page The algebraic and geometric multiple of λ can not determine completely the Jordan structure corresponding to λ

Assume that 1 is an eigenvalue of A and geometric multiple of 1 is 3 algebraic multiple of 1 is 5 then 3 blocks in corresponding to λ the sum of sizes of these blocks is 5 Therefore (see next page)

or

Annihilating polynomial for A In next page we show that A has an annihilating polynomial. Let p(t) be a polynomial. If p(A)=0, then we say p(t) annihilates A and p(t) is an annihilating polynomial for A

Minimal polynomial of A The minimal polynomial of A is monic polynomial of least degree that annihilates A and is denoted by the proof in next page

Caley-Hamilton Theorem This Theorem implies that

Minimal Polynomial when A~B the proof in next page

Mimimal poly. of Jordan matrix Given example to explain in next page

Similarly,

Mimimal poly. of Jordan matrix Proof in next page

index of eigenvalue p.1 See next page

index of eigenvalue p.2

index of eigenvalue p.3

Observation 6 for Jordan matrix p.1 ….

Observation 6 for Jordan matrix p.2 the proof in next page

Observation 7 for Jordan matrix p.2 the proof in next page The number of blocks in of size ≧ k is

See next page 組合矩陣理 論 第一章 Exercise.doc

Jordan structures for The Jordan structure of A corresponding to and that corresponding to are the same. Because

The proof is in next page.

Permutation similarity The proof is in next page

similarly Prove d in next page

similarly

Theorem 1.3.4

By Exercise 組合矩陣理論 第 一 章 Exercise.doc