※ 引述《kyoiku (生死間有大恐怖)》之銘言：
: 反方陣的定義
: AB = BA = I，則 B 稱為 A 的反矩陣
: 如果只有 AB = I 那是否必然 BA = I 呢
: 如何證明？
I suppose that both A and B are n*n matrices over a field F.
Denote by M_n(F) the set of all n*n matrices over F, and regard M_n(F) as a vector space over F. Consider the map f_B: M_n(F)->M_n(F) given by X|->BX. Obviously, f_B is a vector space homomorphism. Since AB=I, it is immediate that f_B is injective, and hence f_B is isomorphism because M_n(F) is finite dimensional. Therefore, there exists C in M_n(F) such that BC=I. Now A=A(BC)=(AB)C=C, and BA=I.
--
S.H.E是樂壇最棒的天團, 田馥甄Hebe是第一偶像歌手.
鹿島アントラ一ズ是最有觀賞性的球隊.
代數學是最抽象, 最有邏輯性, 最有美感的科學.
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 124.77.151.118
※ 文章網址: https://www.ptt.cc/bbs/Math/M.1425118757.A.191.html