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  • 两个公式

    【转载请注明出处】http://www.cnblogs.com/mashiqi

    第一个公式:

    这个公式的证明需要用到

    第二个公式:

    证明如下:

    [egin{array}{l}
    left| {{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}} ight| = left| {egin{array}{*{20}{c}}
    {{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}}&0\
    0&{{I_n}}
    end{array}} ight|\
    mathop {{ m{ = = = = = = = = = = }}}limits^{youchengleft| {egin{array}{*{20}{c}}
    {{I_{p imes p}}}&0\
    {{B_{n imes p}}}&{{I_n}}
    end{array}} ight|} left| {egin{array}{*{20}{c}}
    {{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}}&0\
    {{B_{n imes p}}}&{{I_n}}
    end{array}} ight|\
    mathop {{ m{ = = = = = = = = = = = }}}limits^{zuochengleft| {egin{array}{*{20}{c}}
    {{I_{p imes p}}}&{ - {C_{p imes n}}A_{n imes n}^{ - 1}}\
    0&{{I_n}}
    end{array}} ight|} left| {egin{array}{*{20}{c}}
    {{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}} - {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}}&{ - {C_{p imes n}}A_{n imes n}^{ - 1}}\
    {{B_{n imes p}}}&{{I_n}}
    end{array}} ight|{ m{ = }}left| {egin{array}{*{20}{c}}
    {{I_{p imes p}}}&{ - {C_{p imes n}}A_{n imes n}^{ - 1}}\
    {{B_{n imes p}}}&{{I_n}}
    end{array}} ight|\
    mathop {{ m{ = = = = = = = = = = = }}}limits^{youchengleft| {egin{array}{*{20}{c}}
    {{I_{p imes p}}}&{{C_{p imes n}}A_{n imes n}^{ - 1}}\
    0&{{I_n}}
    end{array}} ight|} left| {egin{array}{*{20}{c}}
    {{I_{p imes p}}}&{{C_{p imes n}}A_{n imes n}^{ - 1} - {C_{p imes n}}A_{n imes n}^{ - 1}}\
    {{B_{n imes p}}}&{{I_n}{ m{ + }}{B_{n imes p}}{C_{p imes n}}A_{n imes n}^{ - 1}}
    end{array}} ight|{ m{ = }}left| {egin{array}{*{20}{c}}
    {{I_{p imes p}}}&0\
    {{B_{n imes p}}}&{{I_n}{ m{ + }}{B_{n imes p}}{C_{p imes n}}A_{n imes n}^{ - 1}}
    end{array}} ight|\
    { m{ = }}left| {egin{array}{*{20}{c}}
    {{I_{p imes p}}}&0\
    {{B_{n imes p}}}&{{I_n}{ m{ + }}{B_{n imes p}}{C_{p imes n}}}
    end{array}} ight| cdot left| {A_{n imes n}^{ - 1}} ight|{ m{ = }}left| {A_{n imes n}^{ - 1}} ight| cdot left| {{A_{n imes n}} + {B_{n imes p}}{C_{p imes n}}} ight|
    end{array}]

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  • 原文地址:https://www.cnblogs.com/mashiqi/p/3717763.html
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