每週問題 March 27, 2017

若對角矩陣有相異對角元與某個矩陣是可交換的,則該矩陣也是對角矩陣。

Prove the following statements.
(a) Let D=\hbox{diag}(d_1,\ldots,d_n), where d_i are distinct. If AD=DA, then A is a diagonal matrix.
(b) Let D=\hbox{diag}(d_1,\ldots,d_n), where d_i are distinct and nonzero, and N=[n_{ij}] be an n\times n matrix, where n_{ij}=\delta_{i+1,j}. If AD=DA and NDA=AND, then A=aI.

 
參考解答:

(a) 令 A=[a_{ij}]n\times n 階矩陣。寫出 (AD)_{ij}=d_ja_{ij}(DA)_{ij}=d_ia_{ij}1\le i,j\le n。因此,d_ja_{ij}=d_ia_{ij}(d_j-d_i)a_{ij}=0。當 i\neq jd_i\neq d_j 推得 a_{ij}=0

(b) 由 (a) 可知 A=[a_{ij}] 是對角矩陣。不難驗證 (NDA)_{i,i+1}=d_{i+1}a_{i+1,i+1}(AND)_{i,i+1}=d_{i+1}a_{ii},因此 a_{ii}=a_{i+1,i+1}i=1,\ldots,n-1

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