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[擂台] 行列式为平方数的平方元素矩阵

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发表于 2019-4-11 20:15:29 | 显示全部楼层 |阅读模式

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平方元素矩阵,顾名思义,就是其元素全都为平方数的矩阵。
可以通过行列互易对角化的矩阵为平凡解。我们要求非平凡解。
进一步,可以拓宽数域到有理数的平方数。
不仅要求非平凡解,甚至可以要求全部元素非零,称为无零解。
二阶易解,三阶的呢?
  1. Det[( {
  2.    {Subscript[a, 11], Subscript[a, 12], Subscript[a, 13]},
  3.    {Subscript[a, 21], Subscript[a, 22], Subscript[a, 23]},
  4.    {Subscript[a, 31], Subscript[a, 32], Subscript[a, 33]}
  5.   } )]
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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2019-4-11 21:14:06 来自手机 | 显示全部楼层
主对角线元素为平方数的三角阵
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2019-4-12 15:23:24 | 显示全部楼层

无零解

1,1,1
1,4,1
1,1,4
结果是9.
其实只要其他的都固定,留一个元素作为未知数x,结果是x的线性函数,解一个二次同余方程就行了。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2019-4-12 16:26:01 来自手机 | 显示全部楼层
所以lsr的方案可以扩展到任意奇数阶无零解
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2019-4-12 16:53:48 来自手机 | 显示全部楼层
第二行前两个数选16,25,第三行开始主对角线为4,所有其它元素为1,对偶数阶都可行
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2023-3-16 10:53:49 | 显示全部楼层

无零解

  1. SL[X_] := Piecewise[{
  2.    {Reverse[Transpose[Reverse[X]]],
  3.     First[Diagonal[X]] > Last[Diagonal[X]]},
  4.    {X, First[Diagonal[X]] < Last[Diagonal[X]]},
  5.    {Piecewise[{
  6.       {Reverse[Transpose[Reverse[X]]],
  7.        First[Diagonal[X, 1]] > Last[Diagonal[X, 1]]},
  8.       {X, First[Diagonal[X, 1]] <= Last[Diagonal[X, 1]]}}],
  9.     First[Diagonal[X]] == Last[Diagonal[X]]}
  10.    }]
  11. Llst = {{{1, 1, 1}, {1, 4, 1}, {1, 1, 4}}, {{1, 1, 1}, {1, 4, 4}, {1,
  12.      4, 16}}, {{1, 1, 1}, {1, 9, 1}, {1, 1, 9}}, {{1, 1, 1}, {1, 9,
  13.      25}, {1, 25, 81}}, {{1, 1, 1}, {1, 16, 4}, {1, 4, 4}}, {{1, 1,
  14.      1}, {1, 81, 25}, {1, 25, 9}}, {{1, 1, 4}, {1, 9, 16}, {4, 16,
  15.      36}}, {{1, 4, 1}, {4, 36, 16}, {1, 16, 9}}, {{1, 4, 4}, {4, 1,
  16.      4}, {4, 4, 1}}, {{1, 4, 4}, {4, 1, 4}, {4, 4, 4}}, {{1, 4,
  17.      4}, {4, 4, 4}, {4, 4, 1}}, {{1, 4, 4}, {4, 9, 16}, {4, 16,
  18.      9}}, {{1, 4, 4}, {4, 25, 4}, {4, 4, 36}}, {{1, 4, 4}, {4, 25,
  19.      16}, {4, 16, 25}}, {{1, 4, 4}, {4, 36, 4}, {4, 4, 25}}, {{1, 9,
  20.      9}, {9, 1, 9}, {9, 9, 16}}, {{1, 9, 9}, {9, 16, 9}, {9, 9,
  21.      1}}, {{1, 9, 9}, {9, 16, 49}, {9, 49, 64}}, {{1, 9, 9}, {9, 64,
  22.      49}, {9, 49, 16}}, {{4, 1, 1}, {1, 1, 1}, {1, 1, 4}}, {{4, 1,
  23.      1}, {1, 4, 1}, {1, 1, 1}}, {{4, 1, 4}, {1, 1, 1}, {4, 1,
  24.      16}}, {{4, 1, 4}, {1, 49, 25}, {4, 25, 16}}, {{4, 4, 1}, {4, 16,
  25.      1}, {1, 1, 1}}, {{4, 4, 1}, {4, 16, 25}, {1, 25, 49}}, {{4, 4,
  26.      4}, {4, 1, 4}, {4, 4, 1}}, {{4, 4, 4}, {4, 9, 1}, {4, 1,
  27.      9}}, {{4, 4, 4}, {4, 9, 16}, {4, 16, 36}}, {{4, 4, 4}, {4, 36,
  28.      16}, {4, 16, 9}}, {{4, 4, 16}, {4, 9, 25}, {16, 25, 81}}, {{4, 9,
  29.       9}, {9, 4, 36}, {9, 36, 4}}, {{4, 9, 36}, {9, 4, 9}, {36, 9,
  30.      4}}, {{4, 16, 4}, {16, 81, 25}, {4, 25, 9}}, {{4, 36, 9}, {36, 4,
  31.       9}, {9, 9, 4}}, {{9, 1, 1}, {1, 1, 1}, {1, 1, 9}}, {{9, 1,
  32.      1}, {1, 9, 1}, {1, 1, 1}}, {{9, 1, 4}, {1, 9, 4}, {4, 4,
  33.      4}}, {{9, 1, 16}, {1, 1, 4}, {16, 4, 36}}, {{9, 1, 25}, {1, 1,
  34.      1}, {25, 1, 81}}, {{9, 4, 1}, {4, 4, 4}, {1, 4, 9}}, {{9, 4,
  35.      16}, {4, 1, 4}, {16, 4, 9}}, {{9, 4, 16}, {4, 4, 4}, {16, 4,
  36.      36}}, {{9, 4, 25}, {4, 4, 16}, {25, 16, 81}}, {{9, 16, 1}, {16,
  37.      36, 4}, {1, 4, 1}}, {{9, 16, 4}, {16, 9, 4}, {4, 4, 1}}, {{9, 16,
  38.       4}, {16, 36, 4}, {4, 4, 4}}, {{9, 16, 16}, {16, 36, 16}, {16,
  39.      16, 49}}, {{9, 16, 16}, {16, 49, 16}, {16, 16, 36}}, {{9, 25,
  40.      1}, {25, 81, 1}, {1, 1, 1}}, {{9, 25, 4}, {25, 81, 16}, {4, 16,
  41.      4}}, {{16, 1, 4}, {1, 1, 1}, {4, 1, 4}}, {{16, 4, 1}, {4, 4,
  42.      1}, {1, 1, 1}}, {{16, 4, 25}, {4, 4, 1}, {25, 1, 49}}, {{16, 9,
  43.      9}, {9, 1, 9}, {9, 9, 1}}, {{16, 9, 49}, {9, 1, 9}, {49, 9,
  44.      64}}, {{16, 25, 4}, {25, 49, 1}, {4, 1, 4}}, {{16, 49, 9}, {49,
  45.      64, 9}, {9, 9, 1}}, {{25, 4, 4}, {4, 1, 4}, {4, 4, 36}}, {{25, 4,
  46.       4}, {4, 36, 4}, {4, 4, 1}}, {{25, 4, 16}, {4, 1, 4}, {16, 4,
  47.      25}}, {{25, 16, 4}, {16, 25, 4}, {4, 4, 1}}, {{36, 4, 4}, {4, 1,
  48.      4}, {4, 4, 25}}, {{36, 4, 4}, {4, 25, 4}, {4, 4, 1}}, {{36, 4,
  49.      16}, {4, 1, 1}, {16, 1, 9}}, {{36, 4, 16}, {4, 4, 4}, {16, 4,
  50.      9}}, {{36, 16, 4}, {16, 9, 1}, {4, 1, 1}}, {{36, 16, 4}, {16, 9,
  51.      4}, {4, 4, 4}}, {{36, 16, 16}, {16, 9, 16}, {16, 16, 49}}, {{36,
  52.      16, 16}, {16, 49, 16}, {16, 16, 9}}, {{49, 1, 25}, {1, 4,
  53.      4}, {25, 4, 16}}, {{49, 16, 16}, {16, 9, 16}, {16, 16,
  54.      36}}, {{49, 16, 16}, {16, 36, 16}, {16, 16, 9}}, {{49, 25,
  55.      1}, {25, 16, 4}, {1, 4, 4}}, {{64, 9, 49}, {9, 1, 9}, {49, 9,
  56.      16}}, {{64, 49, 9}, {49, 16, 9}, {9, 9, 1}}, {{81, 1, 25}, {1, 1,
  57.       1}, {25, 1, 9}}, {{81, 16, 25}, {16, 4, 4}, {25, 4, 9}}, {{81,
  58.      25, 1}, {25, 9, 1}, {1, 1, 1}}, {{81, 25, 16}, {25, 9, 4}, {16,
  59.      4, 4}}};
  60. MatrixForm /@ Llst;
  61. MatrixForm /@ SL /@ DeleteDuplicatesBy[Llst, SL[#] &]
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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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