       • To: mathgroup at smc.vnet.net
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Mon, 29 Oct 2007 05:54:37 -0500 (EST)

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

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,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][]
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.