Question

我有这个矩阵

a = {{2, -2, -4}, {-2, 5, -2}, {-4, -2, 2}}

然后我解决了一个缺少一个条目的等式。等式是形式反[p] .a.p == q 其中p是3x3矩阵，缺少条目（x5），q是给定的3x3矩阵。

Solve[Inverse[( {
      {1/Sqrt[5], 4/(3 Sqrt[5]), -2/3},
      {-2/Sqrt[5], 2/(3 Sqrt[5]), -2/6},
      {0, x5, -2/3}
     } )].a.( {
     {1/Sqrt[5], 4/(3 Sqrt[5]), -2/3},
     {-2/Sqrt[5], 2/(3 Sqrt[5]), -2/6},
     {0, x5, -2/3}
    } ) == ( {
    {6, 0, 0},
    {0, 6, 0},
    {0, 0, -3}
   } )]

Mathematica可以轻松解决这个问题，我得到x5 - ＆gt; - （Sqrt [5] / 3）作为结果。但是，如果我检查它，结果非常奇怪：

In[2]:= Inverse[( {
    {1/Sqrt[5], 4/(3 Sqrt[5]), -2/3},
    {-2/Sqrt[5], 2/(3 Sqrt[5]), -2/6},
    {0, -Sqrt[5]/3, -2/3}
   } )].a.( {
   {1/Sqrt[5], 4/(3 Sqrt[5]), -2/3},
   {-2/Sqrt[5], 2/(3 Sqrt[5]), -2/6},
   {0, -Sqrt[5]/3, -2/3}
  } )

Out[2]= {{6/5 - (2 (-(2/Sqrt[5]) - 2 Sqrt[5]))/Sqrt[5], 
  8/5 + (2 (-(2/Sqrt[5]) - 2 Sqrt[5]))/(3 Sqrt[5]), -(4/Sqrt[5]) + 
   1/3 (2/Sqrt[5] + 2 Sqrt[5])}, {-((
    2 (-(8/(3 Sqrt[5])) + (4 Sqrt[5])/3))/Sqrt[5]) + (
   4/(3 Sqrt[5]) + (4 Sqrt[5])/3)/Sqrt[5], 
  10/3 + (2 (-(8/(3 Sqrt[5])) + (4 Sqrt[5])/3))/(3 Sqrt[5]) + (
   4 (4/(3 Sqrt[5]) + (4 Sqrt[5])/3))/(3 Sqrt[5]), (4 Sqrt[5])/3 + 
   1/3 (8/(3 Sqrt[5]) - (4 Sqrt[5])/3) - 
   2/3 (4/(3 Sqrt[5]) + (4 Sqrt[5])/3)}, {0, 0, -3}}

预期结果应为

( {
  {6, 0, 0},
  {0, 6, 0},
  {0, 0, -3}
 } )

像方程式一样。如果我手动计算，我会得到这个结果。我在这里缺少什么？

Answer 1

结果只需Simplify或Expand。

以下是一个例子：

In[1]:= a = {{2, -2, -4}, {-2, 5, -2}, {-4, -2, 2}}
Out[1]= {{2, -2, -4}, {-2, 5, -2}, {-4, -2, 2}}

In[2]:= p = {{1/Sqrt[5], 4/(3 Sqrt[5]), -(2/3)}, {-(2/Sqrt[5]), 2/(
   3 Sqrt[5]), -(2/6)}, {0, x5, -(2/3)}}

Out[2]= {{1/Sqrt[5], 4/(3 Sqrt[5]), -(2/3)}, {-(2/Sqrt[5]), 2/(
  3 Sqrt[5]), -(1/3)}, {0, x5, -(2/3)}}

In[3]:= sol = 
 Solve[Inverse[p].a.p == {{6, 0, 0}, {0, 6, 0}, {0, 0, -3}}]

Out[3]= {{x5 -> -(Sqrt[5]/3)}}

In[4]:= Inverse[p].a.p /. sol[[1]]
Out[4]= <big output removed>

In[5]:= Simplify[%]
Out[5]= {{6, 0, 0}, {0, 6, 0}, {0, 0, -3}}

Expand也可以取代Simplify。在根和分数方面的表达通常可以用几种方式编写，并且如果两个表达式仅仅通过查看它们就不是很明显。您必须明确要求Mathematica对其进行转换，例如expr = 13/(2 Sqrt[3]) - 4/3和Together[expr]。

虽然有什么奇怪的，但如果您使用标准语法并明确提供变量，则Solve不起作用：

In[6]:= Solve[Inverse[p].a.p == {{6, 0, 0}, {0, 6, 0}, {0, 0, -3}}, x5]

Out[6]= {}

In[7]:= Solve[
 Inverse[p].a.p == {{6, 0, 0}, {0, 6, 0}, {0, 0, -3}}, x5, 
 VerifySolutions -> False]

Out[7]= {}

任何人都可以解释原因吗？ NSolve按预期工作。

In[8]:= NSolve[
 Inverse[p].a.p == {{6, 0, 0}, {0, 6, 0}, {0, 0, -3}}, x5]

Out[8]= {{x5 -> -0.745356}}

Answer 2

Remove["Global`*"];
a = {{2, -2, -4}, {-2, 5, -2}, {-4, -2, 2}};
p = {{1/Sqrt[5], 4/(3 Sqrt[5]), -2/3}, {-2/Sqrt[5], 
    2/(3 Sqrt[5]), -2/6}, {0, x, -2/3}};
pInv = Inverse[p];
lhs = pInv.a.p;

q = {6, 6, -3};
eqs = N@Expand@
   Map[Total[lhs[[#, All]]] - q[[#]] == 0 &, Range[Length[q]]]

以下是x中的3个方程式。（3个等式，ONE未知！）

-6. - 2.66667/(-0.444444 + 0.745356 x) + (4.47214 x)/(-0.444444 + 0.745356 x) == 
  0., 

-6. - 2.66667/(-0.444444 + 0.745356 x) + (4.47214 x)/(-0.444444 + 0.745356 x) == 0., 

 3. - 0.654283/(-0.444444 + 0.745356 x) -(1.5694 x)/(-0.444444 + 0.745356 x) + (
   4.47214 x^2)/(-0.444444 + 0.745356 x) == 0.

首先用数字解决

 Map[NSolve[eqs[[#]],x]&,Range[3]]

 Out[465]= {{{x->0.}},{{x->0.}},{{x->-0.745356}}}

要让Solve接受x，首先不要做数字，留下它的象征性：

eqs = Expand@ Map[Total[lhs[[#, All]]] - q[[#]] == 0 &, Range[Length[q]]]

给出了

{-6 - 8/(3 (-(4/9) + (Sqrt[5] x)/3)) + (2 Sqrt[5] x)/(-(4/9) + (Sqrt[5] x)/3) == 
  0, 

-6 - 8/(3 (-(4/9) + (Sqrt[5] x)/3)) + (2 Sqrt[5] x)/(-(4/9) + (Sqrt[5] x)/3) == 0, 

 3 + 4/(3 (-(4/9) + (Sqrt[5] x)/3)) - (8 Sqrt[5])/(9 (-(4/9) + (Sqrt[5] x)/3)) 
 + (2 x)/(3 (-(4/9) + (Sqrt[5] x)/3)) - (
   Sqrt[5] x)/(-(4/9) + (Sqrt[5] x)/3) + (
   2 Sqrt[5] x^2)/(-(4/9) + (Sqrt[5] x)/3) == 0}

现在使用Solve，其中显式x，现在没问题

Map[Solve[eqs[[#]], x] &, Range[3]]

{{{}}, {{}}, {{x -> -(Sqrt[5]/3)}}}

- 纳赛尔

为什么不回复Solve []的结果给出预期的结果？

2 个答案: