每週問題 July 18, 2016

這是關於反對稱矩陣 (skew symmetric matrix) 與反 Hermitian 矩陣的問題。

Prove that each of the following statements is true.
(a) If A=[a_{ij}] is skew symmetric, then a_{ii}=0 for each i.
(b) If A=[a_{ij}] is skew Hermitian, then each a_{ii} is a pure imaginary number.
(c) If A is real and symmetric, then B=\mathrm{i}A is skew Hermitian, where \mathrm{i}=\sqrt{-1}.

 
參考解答:

(a) 若 A^T=-A,則 a_{ji}=-a_{ij}。當 i=ja_{ii}=-a_{ii},故 a_{ii}=0

(b) 若 A^\ast=-A,則 \overline{a_{ji}}=-a_{ij}。當 i=j\overline{a_{ii}}=-a_{ii}。寫出 a_{ii}=a+\mathrm{i}b,其中 \mathrm{i}=\sqrt{-1},即有 a-\mathrm{i}b=-a-\mathrm{i}b,因此 a=0,故得證。

(c) 因為 \overline{A}=AA^T=A,可得

B^\ast=(\mathrm{i}A)^\ast=-\mathrm{i}A^\ast=-\mathrm{i}\overline{A}^T=-\mathrm{i}A^T=-\mathrm{i}A=-B

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