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