葡萄糖 发表于 2019-4-11 20:15:29

行列式为平方数的平方元素矩阵

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？:tip:
Det[( {
   {Subscript, Subscript, Subscript},
   {Subscript, Subscript, Subscript},
   {Subscript, Subscript, Subscript}
} )]

mathe 发表于 2019-4-11 21:14:06

主对角线元素为平方数的三角阵

lsr314 发表于 2019-4-12 15:23:24

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。

mathe 发表于 2019-4-12 16:26:01

所以lsr的方案可以扩展到任意奇数阶无零解

mathe 发表于 2019-4-12 16:53:48

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

葡萄糖 发表于 2023-3-16 10:53:49

无零解

SL := Piecewise[{
   {Reverse]],
    First] > Last]},
   {X, First] < Last]},
   {Piecewise[{
      {Reverse]],
       First] > Last]},
      {X, First] <= Last]}}],
    First] == Last]}
   }]
Llst = {{{1, 1, 1}, {1, 4, 1}, {1, 1, 4}}, {{1, 1, 1}, {1, 4, 4}, {1,
   4, 16}}, {{1, 1, 1}, {1, 9, 1}, {1, 1, 9}}, {{1, 1, 1}, {1, 9,
   25}, {1, 25, 81}}, {{1, 1, 1}, {1, 16, 4}, {1, 4, 4}}, {{1, 1,
   1}, {1, 81, 25}, {1, 25, 9}}, {{1, 1, 4}, {1, 9, 16}, {4, 16,
   36}}, {{1, 4, 1}, {4, 36, 16}, {1, 16, 9}}, {{1, 4, 4}, {4, 1,
   4}, {4, 4, 1}}, {{1, 4, 4}, {4, 1, 4}, {4, 4, 4}}, {{1, 4,
   4}, {4, 4, 4}, {4, 4, 1}}, {{1, 4, 4}, {4, 9, 16}, {4, 16,
   9}}, {{1, 4, 4}, {4, 25, 4}, {4, 4, 36}}, {{1, 4, 4}, {4, 25,
   16}, {4, 16, 25}}, {{1, 4, 4}, {4, 36, 4}, {4, 4, 25}}, {{1, 9,
   9}, {9, 1, 9}, {9, 9, 16}}, {{1, 9, 9}, {9, 16, 9}, {9, 9,
   1}}, {{1, 9, 9}, {9, 16, 49}, {9, 49, 64}}, {{1, 9, 9}, {9, 64,
   49}, {9, 49, 16}}, {{4, 1, 1}, {1, 1, 1}, {1, 1, 4}}, {{4, 1,
   1}, {1, 4, 1}, {1, 1, 1}}, {{4, 1, 4}, {1, 1, 1}, {4, 1,
   16}}, {{4, 1, 4}, {1, 49, 25}, {4, 25, 16}}, {{4, 4, 1}, {4, 16,
   1}, {1, 1, 1}}, {{4, 4, 1}, {4, 16, 25}, {1, 25, 49}}, {{4, 4,
   4}, {4, 1, 4}, {4, 4, 1}}, {{4, 4, 4}, {4, 9, 1}, {4, 1,
   9}}, {{4, 4, 4}, {4, 9, 16}, {4, 16, 36}}, {{4, 4, 4}, {4, 36,
   16}, {4, 16, 9}}, {{4, 4, 16}, {4, 9, 25}, {16, 25, 81}}, {{4, 9,
      9}, {9, 4, 36}, {9, 36, 4}}, {{4, 9, 36}, {9, 4, 9}, {36, 9,
   4}}, {{4, 16, 4}, {16, 81, 25}, {4, 25, 9}}, {{4, 36, 9}, {36, 4,
      9}, {9, 9, 4}}, {{9, 1, 1}, {1, 1, 1}, {1, 1, 9}}, {{9, 1,
   1}, {1, 9, 1}, {1, 1, 1}}, {{9, 1, 4}, {1, 9, 4}, {4, 4,
   4}}, {{9, 1, 16}, {1, 1, 4}, {16, 4, 36}}, {{9, 1, 25}, {1, 1,
   1}, {25, 1, 81}}, {{9, 4, 1}, {4, 4, 4}, {1, 4, 9}}, {{9, 4,
   16}, {4, 1, 4}, {16, 4, 9}}, {{9, 4, 16}, {4, 4, 4}, {16, 4,
   36}}, {{9, 4, 25}, {4, 4, 16}, {25, 16, 81}}, {{9, 16, 1}, {16,
   36, 4}, {1, 4, 1}}, {{9, 16, 4}, {16, 9, 4}, {4, 4, 1}}, {{9, 16,
      4}, {16, 36, 4}, {4, 4, 4}}, {{9, 16, 16}, {16, 36, 16}, {16,
   16, 49}}, {{9, 16, 16}, {16, 49, 16}, {16, 16, 36}}, {{9, 25,
   1}, {25, 81, 1}, {1, 1, 1}}, {{9, 25, 4}, {25, 81, 16}, {4, 16,
   4}}, {{16, 1, 4}, {1, 1, 1}, {4, 1, 4}}, {{16, 4, 1}, {4, 4,
   1}, {1, 1, 1}}, {{16, 4, 25}, {4, 4, 1}, {25, 1, 49}}, {{16, 9,
   9}, {9, 1, 9}, {9, 9, 1}}, {{16, 9, 49}, {9, 1, 9}, {49, 9,
   64}}, {{16, 25, 4}, {25, 49, 1}, {4, 1, 4}}, {{16, 49, 9}, {49,
   64, 9}, {9, 9, 1}}, {{25, 4, 4}, {4, 1, 4}, {4, 4, 36}}, {{25, 4,
      4}, {4, 36, 4}, {4, 4, 1}}, {{25, 4, 16}, {4, 1, 4}, {16, 4,
   25}}, {{25, 16, 4}, {16, 25, 4}, {4, 4, 1}}, {{36, 4, 4}, {4, 1,
   4}, {4, 4, 25}}, {{36, 4, 4}, {4, 25, 4}, {4, 4, 1}}, {{36, 4,
   16}, {4, 1, 1}, {16, 1, 9}}, {{36, 4, 16}, {4, 4, 4}, {16, 4,
   9}}, {{36, 16, 4}, {16, 9, 1}, {4, 1, 1}}, {{36, 16, 4}, {16, 9,
   4}, {4, 4, 4}}, {{36, 16, 16}, {16, 9, 16}, {16, 16, 49}}, {{36,
   16, 16}, {16, 49, 16}, {16, 16, 9}}, {{49, 1, 25}, {1, 4,
   4}, {25, 4, 16}}, {{49, 16, 16}, {16, 9, 16}, {16, 16,
   36}}, {{49, 16, 16}, {16, 36, 16}, {16, 16, 9}}, {{49, 25,
   1}, {25, 16, 4}, {1, 4, 4}}, {{64, 9, 49}, {9, 1, 9}, {49, 9,
   16}}, {{64, 49, 9}, {49, 16, 9}, {9, 9, 1}}, {{81, 1, 25}, {1, 1,
      1}, {25, 1, 9}}, {{81, 16, 25}, {16, 4, 4}, {25, 4, 9}}, {{81,
   25, 1}, {25, 9, 1}, {1, 1, 1}}, {{81, 25, 16}, {25, 9, 4}, {16,
   4, 4}}};
MatrixForm /@ Llst;
MatrixForm /@ SL /@ DeleteDuplicatesBy &]
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