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Calculate generalized eigenvector using Jordan normal form.

Posted 4 months ago
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I try to reproduce the example 5 given here:

enter image description here

In[173]:= (*Example 5*)
mA={{-14,2,-2,0},{4,0,2,0},{-3,0,0,1},{1,0,0,0}}
mA//JordanDecomposition//RootReduce

Out[173]= {{-14, 2, -2, 0}, {4, 0, 2, 0}, {-3, 0, 0, 1}, {1, 0, 0, 0}}

Out[174]= {{{Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 1, 0], 
   Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 2, 0], 
   Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 3, 0], 
   Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 4, 0]}, {Root[
   462 - 274 # + 48 #^2 - 11 #^3 + 2 #^4& , 2, 0], Root[
   462 - 274 # + 48 #^2 - 11 #^3 + 2 #^4& , 1, 0], Root[
   462 - 274 # + 48 #^2 - 11 #^3 + 2 #^4& , 3, 0], Root[
   462 - 274 # + 48 #^2 - 11 #^3 + 2 #^4& , 4, 0]}, {Root[
   29 + 124 # + 104 #^2 + 34 #^3 + 4 #^4& , 1, 0], Root[
   29 + 124 # + 104 #^2 + 34 #^3 + 4 #^4& , 2, 0], Root[
   29 + 124 # + 104 #^2 + 34 #^3 + 4 #^4& , 4, 0], Root[
   29 + 124 # + 104 #^2 + 34 #^3 + 4 #^4& , 3, 0]}, {1, 1, 1, 
   1}}, {{Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 1, 0], 0, 0, 
   0}, {0, Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 2, 0], 0, 0}, {0,
    0, Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 3, 0], 0}, {0, 0, 0, 
   Root[-4 + 14 # - 14 #^2 + 14 #^3 + #^4& , 4, 0]}}}

As shown above, I got a very complicated result and couldn't get the simple form given in the Wiki. Any tips for solving this problem?

Regards,
Zhao

POSTED BY: Hongyi Zhao
2 Replies

You might want to take the Jordan decomposition of A rather than the conversion matrix M. Just a suggestion.

POSTED BY: Daniel Lichtblau
Posted 4 months ago

Thank you for pointing this out. I rechecked as follows based on your suggestion:

In[181]:= A={{5,1,-2,4},{0,5,2,2},{0,0,5,3},{0,0,0,4}};
{M0,J}=A//JordanDecomposition
A . M0==M0 . J
(*
Here is the M matrix given in Example 5 on Wikipedia
*)
M={{-14, 2, -2, 0}, {4, 0, 2, 0}, {-3, 0, 0, 1}, {1, 0, 0, 0}};
A . M==M . J

Out[182]= {{{-14, 1, 0, 0}, {4, 0, 1, 1}, {-3, 0, 0, 1/2}, {1, 0, 0, 
   0}}, {{4, 0, 0, 0}, {0, 5, 1, 0}, {0, 0, 5, 1}, {0, 0, 0, 5}}}

Out[183]= True

Out[185]= True

As you can see, both $M$ and $M_0$ matrices are the valid canonical bases of linearly independent generalized eigenvectors for matrix ${\displaystyle A}$.

There are countless choices for such bases, but what is the relationship between these canonical bases meeting this condition?

POSTED BY: Hongyi Zhao
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