補充一下反方陣的靈感來源,不然直接蹦出那麼複雜的方陣,不是很直觀。下面用英文寫:
Since
Im−AB is invertible, there exists an
m
m matrix
C such that
(Im−AB)C=C(Im−AB)=Im,
hence,
ABC=CAB.
So,
In−BA=In−B(Im−AB)CA=In−BCA+B(ABC)A=In−BCA+B(CAB)A=In−(BCA−BCABA)=In−BCA(In−BA),
therefore,
In−BA=In−BCA(In−BA).
This implies
(In−BA)+BCA(In−BA)=In, so we get
(In+BCA)(In−BA)=In.
Finally, check
(In−BA)(In+BCA)=In to get the conslusion.
Furthermore, substitute
C with
(Im−AB)−1, we get
(In−BA)−1=(In+B(Im−AB)−1A)
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本帖最後由 swallow7103 於 2024-7-1 23:19 編輯 ]