- #1
LCSphysicist
- 646
- 162
- Homework Statement
- If (u,v,w) is a positive basis, so (u^v, v^w,w^u) is too.
- Relevant Equations
- All below
I think we can say that (u,v,u^v) is a positive basis, so as (w^v,v,w) and (u,w^u,w). (1)
So
u^v = βw
v^w = γu
w^u = λv
where λ, β, and γ > 0 (*)
(u^v, v^w,w^u) = (βw,γu,λv)
\begin{vmatrix}
0 & 0 & β \\
γ & 0 & 0 \\
0 & λ & 0 \\
\end{vmatrix}
This determinant is positive by (*)
What you think about?
So
u^v = βw
v^w = γu
w^u = λv
where λ, β, and γ > 0 (*)
(u^v, v^w,w^u) = (βw,γu,λv)
\begin{vmatrix}
0 & 0 & β \\
γ & 0 & 0 \\
0 & λ & 0 \\
\end{vmatrix}
This determinant is positive by (*)
What you think about?