[Mathematical Analysis for small readers] (a sufficient condition for symmetric matrix or anti-symmetric 2014-8 8 8)

Set $ A \ in M _ {n} (\ mathbb f) $ to any $ \ Alpha, \ beta \ In \ mathbb f ^ N $ \ Alpha ^ ta \ Beta = 0 \ leftrightarrow \ beta ^ ta \ alpha = 0 $ and $ A $ is not a symmetric matrix, prove $ A ^ t =-A $.

proof: [from Longfeng chengxiang] Only instructions $ A _ {II} = 0 $ and $ A _ {IJ} =-A _ {IJ }, due to asymmetry, I \ NEQ J $ is not general. If you set $ A _ {12} \ NEQ A _ {21} $, the two are not all zero, set $ A _ {12} \ neq0 $. $ A _ {11} = 0 $, otherwise, $ \ beex \ Bea E _ {1} ^ Tae _ {2}-\ frac {A _ {12} {A _ {11} e _ {1} ^ Tae _ {1} & = E _ {1} ^ ta \ left (E _ {2}-\ frac {A _ {12} {A _ {11 }} E _ {1} \ right) = 0 \ rightarrow \ left (E_2-\ frac {A _ {12 }}{ A _ {11} e _ {1} \ right) ^ Tae _ {1} & = 0 \ rightarrow A _ {12} & = A _ {21} \ EEA \ eeex $ conflict! So $ A _ {11} = 0 $. for example, $ A _ {22} = 0 $. besides, $ A _ {12} =-A _ {21} $, note that $ \ beex \ Bea 0 & = A _ {11} A _ {12} + A _ {12} A _ {21}-A _ {21} _ {12}-A _ {22} A _ {21} \ & = \ left (A _ {21} e _ {1}-A _ {12} e _{ 2} \ right) ^ TA (E _ {1} + E _ {2}) \ rightarrow (E _ {1} + E _ {2 }) ^ ta \ left (A _ {21} e _ {1}-A _ {12} e _ {2} \ right) & = 0 \\\ rightarrow A _ {12} ^ 2 & = A _ {21} ^ 2 \ EEA \ eeex $ and the two are different, so $ A _ {12} =-A _ {21} \ neq0 $. then, it must be $ A _ {1j} =-A _ {J1}, j = 3, \ cdots, N $, if $ A _ {1i} = A _ {I1} = 0 $, it is valid. therefore, you only need to consider the situation where $ J $ makes $ A _ {1j} \ neq0 $. then $ \ beex \ Bea 0 & = E _ {1} ^ Tae _ {2}-\ frac {A _ {12} {A _ {1j} e _ {1} ^ Tae _ {J} \ & = E _ {1} ^ ta \ left (E _ {2}-\ frac {A _ {12} {_ {1j }}e _ {J} \ right) \\\ rightarrow \ left (E _ {2}-\ frac {A _ {12 }}{ A _ {1j} e _ {J} \ right) ^ Tae _ {1} & = A _ {21}-\ frac {A _ {12} {A _ {1j} A _ {J1} = 0 \\\ rightarrow A _ {1j} & =-A _ {J1} \ EEA \ eeex $ continue to repeat the above steps to describe $ A $ objection.

Note: [from torsor] This is a theorem in the high-end textbook. You can refer to the theorem 10.3.1 of the second edition of Fudan high-end textbook. If there is no Fudan textbook, you can refer to Roman's advanced linear algebra, 3rd ed. theorem 266th on page 1.

[Mathematical Analysis for small readers] (a sufficient condition for symmetric matrix or anti-symmetric 2014-8 8 8)