每週問題 December 12 , 2016

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 $k\ge 1$,

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

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

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

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