本論文探討「可分解友矩陣」的一些性質。我們證明若一個非么正友矩陣 A 可分解成A1 ⊕ A2，則 且
。令*1 1 rank( ) 1 kI −AA =* =
2 2 rank( ) 1 n k I AA − − { : rank( * )=1 and | | , ( )} n n Sα ≡ A∈M I−AA λ =α ∀λ ∈σ A
，則相當於1 是屬於A k Sα 且2 是屬於A 1/
n k S α
− 。亦證明每一個屬於n Sα
集合內的矩陣均具有循環、不可分解、且其數值域之邊界為一
可微曲線。並證明下列敘述互為等價：(a) ；(b)
；(c)
1 W(A)=W(A)
1 1) n 2 n1 W(J ) W(A − ⊆ W(A ) W(J ) − ⊆ 。

In this thesis, we study some properties of reducible companion matrices. We first prove that if a nonunitary reducible companion matrix A is unitarily equivalent to the direct sum A_1oplus A_2 on mathbb{C}^koplusmathbb{C}^{n-k} with sigma(A_1)={aom_n^{j_1},cdots,aom_n^{j_k}} and sigma(A_2)={(1/ ar{a})om_n^{j_{k+1}},cdots,(1/ ar{a})om_n^{j_n}},where |a|>1 and om_n denotes the nth primitive root of 1,then rank(I_k-A^{*}_1A_1)=rank(I_{n-k}-A^{*}_2A_2)=1. We denote mathcal{S}^{al}_nequiv{Ain M_n:rank(I_n-A^{*}A)=1 and |la|=al,forall: lainsigma(A)}, thus A_1 is in mathcal{S}^{al}_k and A_2 is in mathcal{S}^{1/al}_{n-k}. Next, we prove that every mathcal{S}^{al}_n-matrix is irreducible, cyclic, and the boundary of its numerical range is a differentiable curve.
Furthermore, we show that the following statements are equivalent:
(a) W(A)=W(A_1); (b)W(J_{n-1})subseteq W(A_1); (c)W(A_2)subseteq W(J_{n-1}).

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