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Re: wrong answer or no answer?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82775] Re: wrong answer or no answer?
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 30 Oct 2007 05:49:19 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <fg4e5h$6oa$1@smc.vnet.net>

Roger Bagula wrote:

> M = {{-1, I}, {I, 1}}
> MatrixPower[M, 1/2]
> gives
> {{0, 0}, {0, 0}}

Seems to be an erroneous result (and surprising since the null matrix 
squared is the null matrix itself).

> So try it as {{a,b},{c,d}} squared:
> c = b; a = I*Sqrt[1 + b^2]; d = Sqrt[1 - b^2
> FullSimplify[{a^2 + b c + 1 == 0, a b + b d - I == 0, a c +
>          c d - I == 0, b c + d^2 - 1 == 0}]
> {True, -I + b*Sqrt[1 - b^2] +I* b*Sqrt[1 +
> b^2)]== 0, -I + b*Sqrt[1 - b^2] +I* b*Sqrt[1 +
> b^2)]== 0, True}
> Solve[-I + b*Sqrt[1 - b^2] +I* b*Sqrt[1 +
> b^2)]== 0,b]
> {}
> which says there is no solution.
> 
> This problem comes from the graph of SU(2) and of U(1)*SU(2)
> as a two vertex 3 directed connections (i,j,k) and a 2 vertex with 4 
> directed connections (1,i,j,k}.
> Basically there is either a solution or there is none.
> Mathematica gives zero and the null set from two different approaches.
> {{0,i+k},{j,0}} and {{0,i+k},{j+IdentityMatrix[2],0}}
> 
> It is pretty much a break down of mathematical definitions.
> The matrix M does appear to have not one, but four solutions
> the way I do it:
> 
> M2={{+/-I*Sqrt[1 + b^2], b}, {b, +/-Sqrt[1 + b^2]}}
> I really may be doing it all wrong.
> b=+/-Sqrt[+/-1/2+I/2]
> which gives the stange answers from this code:
> Clear[b]
> M2 = {{I*Sqrt[1 + b^2], b}, {b, Sqrt[1 + b^2]}}
> Det[M2]
> Solve[Det[M2] == 0, b]
> b0 = b /. Solve[Det[M2] == 0, b][[2]]
> M20 = {{-I*Sqrt[1 + b0^2], b0}, {b0, Sqrt[1 + b0^2]}}
> FullSimplify[M20]
> FullSimplify[M20.M20]
> 
> All this leaves me really puzzled.
> Usually Mathematica takes away my doubts,
> but here it isn't any help at all.
> Maybe it is a paradox?

I have not given a lot of thought to your mathematical approach, but 
such a matrix does not seem to exist anyway.

In[1]:= M = {{a, b}, {c, d}}
m1 = MatrixPower[M, 2] // Flatten
m2 = {{-1, I}, {I, 1}} // Flatten

Out[1]= {{a, b}, {c, d}}

Out[2]= {a^2 + b c, a b + b d, a c + c d, b c + d^2}

Out[3]= {-1, \[ImaginaryI], \[ImaginaryI], 1}

In[4]:= Thread[m1 == m2]

Out[4]= {a^2 + b c == -1, a b + b d == \[ImaginaryI],
  a c + c d == \[ImaginaryI], b c + d^2 == 1}

In[5]:= Reduce[%, {a, b, c, d}]

Out[5]= False

Regards,
-- 
Jean-Marc


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