实反对称矩阵的特征值只能为零或纯虚数怎么证？ 怎么证明反对称矩阵的特征值只能是令或纯虚数？！
\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u53ea\u80fd\u4e3a\u96f6\u6216\u7eaf\u865a\u6570\u600e\u4e48\u8bc1\u8bbeA\u4e3an\u9636\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0cr\u4e3aA\u7684\u7279\u5f81\u503c\uff0cx\u4e3aA\u5bf9\u5e94r\u7684\u7279\u5f81\u5217\u5411\u91cf
A*x=r*x
(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*A*x=r*(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*x\u2026\u2026\u2460
\u56e0\u4e3ax\u975e\u96f6\uff0c\u6240\u4ee5(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*x\u662f\u4e00\u4e2a\u6b63\u6570\uff0c\u8bb0\u4e3aX
\u5c06\u2460\u5f0f\u4e24\u8fb9\u5206\u522b\u4f5c\u5171\u8f6d\u8f6c\u7f6e\uff0c\u56e0\u4e3aA\u5b9e\u53cd\u5bf9\u79f0\uff0c\u6240\u4ee5A\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635=-A
(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*(-A)*x=(r\u7684\u5171\u8f6d)*X
-(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*A*x=(r\u7684\u5171\u8f6d)*X\u2026\u2026\u2461
\u5c06\u2460\u2461\u4e24\u5f0f\u76f8\u52a0\uff0c (r+r\u7684\u5171\u8f6d)*X=0
\u56e0\u4e3aX>0\uff0c\u6240\u4ee5r+r\u7684\u5171\u8f6d=0
\u5373r=0\u6216r\u662f\u7eaf\u865a\u6570
\u6269\u5c55\u8d44\u6599\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u6709\u5982\u4e0b\u6027\u8d28\uff1a
\u6027\u8d281\uff1a\u5947\u6570\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u884c\u5217\u5f0f\u503c\u4e3a0\u3002
\u6027\u8d282\uff1a\u5f53A\u4e3an\u9636\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u65f6\uff0c \u6709XTAX =0\u3002
\u6027\u8d283\uff1a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u662f\u96f6\u6216\u7eaf\u865a\u6570\u3002
\u6027\u8d284\uff1a\u82e5A\u4e3a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0cA\u7684\u7279\u5f81\u503c\u03bb= bi(b\u22600)\u6240\u5bf9\u5e94\u7279\u5f81\u5411\u91cf\u03b1+\u03b2i\u4e2d\u5b9e\u90e8\u4e0e\u865a\u90e8\u5bf9\u5e94\u7684\u5411\u91cf\u03b1\u3001\u03b2\u76f8\u4e92\u6b63\u4ea4\u3002
\u6027\u8d285\uff1a\u82e5A\u4e3an\u9636\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u5219\u5b58\u5728n\u9636\u6b63\u4ea4\u77e9\u9635\u0393\u3002
\u8fd9\u4e2a\u4e0d\u96be.\u53cd\u5bf9\u79f0\u77e9\u9635A,\u6ee1\u8db3A'=-A,\u8bbea\u4e3aA\u7684\u7279\u5f81\u503c,x\u4e3a\u5bf9\u5e94\u7279\u5f81\u5411\u91cf.\u5219\u662fAx=ax.
\u5bf9\u4efb\u4e00\u5411\u91cf\u90fd\u6709x'Ax=0(\u56e0\u4e3ax'Ax\u662f\u4e00\u4e2a\u6570,\u6570\u7684\u8f6c\u7f6e\u662f\u5b83\u672c\u8eab,\u5c31\u6709x'Ax=(x'Ax)'=x'A'x=-x'Ax,\u770b\u7b49\u5f0f\u4e24\u8fb9),\u5c24\u5176x\u4e3a\u7279\u5f81\u5411\u91cf\u65f6\u4e5f\u6210\u7acb,\u5219ax'x=x'Ax=0.\u5176\u4e2dx\u4e3a\u975e\u96f6\u5411\u91cf.
\u540c\u7406A\u7684\u5171\u8f6d\u4e5f\u662f\u53cd\u5bf9\u79f0\u9635,\u4e14\u7279\u5f81\u503c\u4e3aa\u5171\u8f6d,\u5bf9\u5e94\u7279\u5f81\u5411\u91cf\u4e3ax\u5171\u8f6d,\u5c31\u6709a\u5171\u8f6dx'\u5171\u8f6dx\u5171\u8f6d=0
\u7531ax'x=0,\u5219a\u4e3a0,\u6216\u7eaf\u865a\u6570(\u8fd9\u8981\u8003\u8651x\u4e3a\u590d\u5411\u91cf\u65f6,x'x\u7684\u60c5\u51b5\u624d\u80fd\u5f97\u51fa\u7ed3\u8bba).
That is, Aα=λα （α=(a1,a2,...,an)'）
we multply by （α共轭）’on both sides
(α共轭)'Aα=(α共轭)'λα=λ(α共轭)'α
on the other hand
(α共轭)'Aα=(α共轭)'(-A')α=-(Aα的共轭)'α=-(λα共轭)'α
so λ（α共轭）'α=-(λα共轭)'α=-λ（α共轭）'α
so λ=-λ
we suppose λ=a+bi
that is a=0
λ=0 or λ=bi
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