■サマーヴィルの等面四面体(その26)

 (その24)(その25)の続き.

 Dii=0,Dij=Dji=dij^2=(j−i)(n+1+i−j),i<j

 Dix=ΣDij

α^2A=|D00,D01,D02,D03,α|

    |D10,D11,D12,D13,α|

    |D20,D21,D22,D23,α|

    |D30,D31,D32,D33,α|

    | α, α, α, α,0|

 まず,第1行を他の行から引いて

|D00,D01   ,D02   ,D03   ,α|

|D10,D11−D01,D12−D02,D13−D03,0|

|D20,D21−D01,D22−D02,D23−D03,0|

|D30,D31−D01,D32−D02,D33−D03,0|

| α, α−D01, α−D02, α−D03,−α|

さらに第2列〜第n列を第1列に加えれば

|D00+D0x+ α,D01   ,D02   ,D03   ,α|

|D10+D1x−D0x,D11−D01,D12−D02,D13−D03,0|

|D20+D2x−D0x,D21−D01,D22−D02,D23−D03,0|

|D30+D3x−D0x,D31−D01,D32−D02,D33−D03,0|

|   3α−D0x, α−D01, α−D02, α−D03,−α|

D00+D0x=D0x,D10+D1x=D0x,D20+D2x=D0x,D30+D3x=D0xα=D0x/3とおけば

−α^2|D11−D01,D12−D02,D13−D03|

   |D21−D01,D22−D02,D23−D03|

   |D31−D01,D32−D02,D33−D03|

E33=−D03+(D13−D03)(D31−D01)/D01

−α^2|D11−D01,D12−D02,0|

   |D21−D01,D22−D02,D23−D03|

   |    0,D32−D02,E33|

E22=−D02+(D12−D02)(D21−D01)/D01=−D02

Dij=Dji=dij^2=(j−i)(n+1+i−j),i<jより,(D12−D02)(D21−D01)=0

−α^2|D11−D01,0,0|

   |    0,E22,D23−D03|

   |    0,D32−D02,E33|

F33=E33−(D23−D03)(D32−D02)/E22

α^2|D01,0,0|

  |0,E22,0|

  |0,0,F33|

E22F33=E22E33−(D23−D03)(D32−D02)

=−D02{−D03+(D13−D03)(D31−D01)/D01}−(D23−D03)(D32−D02)

D01E22F33

=−D02{−D01D03+(D13−D03)(D31−D01)}−D01(D23−D03)(D32−D02)

=D01D02D03−D02(D13−D03)(D31−D01)−D01(D23−D03)(D32−D02)

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D01=D12=D23

D02=D13

D03=D01

D01E22F33

=−D02{−D01D03+(D13−D03)(D31−D01)}−D01(D23−D03)(D32−D02)

=D01D02D03−D02(D02−D03)(D02−D01)−D01(D01−D03)(D01−D02)

=D01D02D03−D02(D02−D03)(D02−D01)

=D01D02D01−D02(D02−D01)(D02−D01)

=D01^2D02−D02(D02−D01)^2

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