数学のブログ

行列と双線形写像 対称作用素 交代行列の対角要素、零

ラング線形代数学(下) (ちくま学芸文庫) (S.ラング(著)、芹沢 正三(翻訳)、筑摩書房)の8章(行列と双線形写像)、3(対称作用素)、練習問題4の解答を求めてみる。

Aを交代行列（歪対称行列）とする。

定義より、

A T = - A a j i = - a i j

よって、

a i i = - a i i a i i = 0

ゆえに、 交代行列の対角要素は0に等しい。

（証明終）

コード(Wolfram Language, Jupyter)

m = {{a, b}, {c, d}}

{{a, b}, {c, d}}
Solve[Transpose[m] == -m]

Equations may not give solutions for all "solve" variables.: Equations may not give solutions for all "solve" variables.
Output
m = {{a, b, c}, {d, e, f}, {g, h, i}}

{{a, b, c}, {d, e, f}, {g, h, i}}
TraditionalForm[%]
Output
Solve[Transpose[m] == -m]

Equations may not give solutions for all "solve" variables.: Equations may not give solutions for all "solve" variables.



Equations may not give solutions for all "solve" variables.: Equations may not give solutions for all "solve" variables.



Equations may not give solutions for all "solve" variables.: Equations may not give solutions for all "solve" variables.



Further output of `1` will be suppressed during this calculation.: Further output of Solve::svars will be suppressed during this calculation.
Output