åè:
设Aæ¯ç§©ä¸º1çné¶æ¹éµ, å
1. Aå¯è¡¨ç¤ºä¸ºÎ±Î²^T, å
¶ä¸Î±,β为nç»´éé¶ååé(æβ^Tαâ 0).
åä¹,è¥A=αβ^T,å
¶ä¸Î±,β为nç»´éé¶ååé(æβ^Tαâ 0), år(A)=1.
2. A^k = (β^Tα)^(k-1)A
3. Açç¹å¾å¼ä¸º α^Tβ(=β^Tα),0,0,...,0
4. tr(A)=α^Tβ
说æ:
1. æ¹æ³: åAçä¸ä¸ªéé¶çè¡åé,设为 β^T,
åå
¶ä½åè¡æ¯æ¤è¡çkiå.
令α=(k1,...,kn)^T, å A=αβ^T.
åä¹, è¥A=αβ^T, å
¶ä¸Î±,β为nç»´éé¶ååé(æβ^Tαâ 0)
å Aâ 0, æ以 r(A)>=1
åå 为 r(A)=r(αβ^T)<=r(α)=1
æ以 r(A)=1.
2. A^k=(αβ^T)(αβ^T)(αβ^T)...(αβ^T)
= α(β^Tα)(β^Tα)(β^T...α)β^T
= (β^Tα)^(k-1)αβ^T
= (β^Tα)^(k-1)A
3.
å 为 Aα=(αβ^T)α=α(β^Tα)=(β^Tα)α
æ以αæ¯Açå±äºç¹å¾å¼Î²^Tα(â 0)çç¹å¾åé
å 为r(A)=1
æ以é½æ¬¡çº¿æ§æ¹ç¨ç»Ax=0çåºç¡è§£ç³»å« n-1 个åé
å³Açå±äºç¹å¾å¼0ç线æ§æ å
³çç¹å¾åéæn-1个
æ以0è³å°æ¯Açn-1éç¹å¾å¼
èné¶æ¹éµæn个ç¹å¾å¼
æ以Açç¹å¾å¼ä¸º β^Tα,0,0,...,0(n-1é)
å±äºç¹å¾å¼0çç¹å¾åé:
设β=(b1,b2,...,bn)^Tâ 0, ä¸å¦¨è®¾b1â 0
åAç»åçè¡åæ¢å为
b1 b2...bn
0 0 ... 0
... ...
0 0 ... 0
Ax=0çåºç¡è§£ç³»ä¸º
(b2,-b1,0,...,0)^T
(b3,0,-b1,...,0)^T
...
(bn,0,0,...,-b1)^T
æ¤å³ä¸ºAçå±äºç¹å¾å¼0çn-1个线æ§æ å
³çç¹å¾åé
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