每週問題 December 12 , 2016

這是關於半幻方 (semi-magic) 矩陣的分解式問題。

An n\times n matrix is said to be a semi-magic matrix if the sums of the rows and columns are all equal. Show that a semi-magic matrix A can be decomposed as A=B+C such that for integer k\ge 1,

A^k=B^k+C^k.

 
參考解答:

B=AC=0 即有平凡分解式。底下考慮非平凡分解式。假設 n\times n 階半幻方矩陣 A=[a_{ij}] 滿足 \sum_{i=1}^na_{ij}=\sum_{j=1}^na_{ij}=s1\le i,j\le n。令 n\times n 階矩陣 E 的所有元皆為 1。因此,EA=AE=sEE^2=nE。令

\displaystyle B=\frac{s}{n}E,~~C=A-\frac{s}{n}E

使用上面的關係式,

\displaystyle BC=\left(\frac{s}{n}E\right)\left(A-\frac{s}{n}E\right)=\frac{s}{n}EA-\frac{s^2}{n^2}E^2=\frac{s}{n}(sE)-\frac{s^2}{n^2}(nE)=0

同樣地,CB=0。利用歸納法即可證明 A^k=(B+C)^k=B^k+C^kk\ge 1

附註:若 s=0,上述建構法給出 B=0C=A;若 A=E,則 B=AC=0。在這兩種情況下,我不確定是否存在非平凡分解式。

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