Ok, I am trying to solve an equation involving matrices (well, tensors actually), which is of the form: $\mathbf{e}^{T} \cdot \mathbf{M} \cdot \mathbf{e}$ = $\mathbf{N}$, where $\mathbf{e}$ is an unknown matrix and both $\mathbf{M}$ and $\mathbf{N}$ are known ($\mathbf{M}$ contains variables).
Essentially, I am trying to find the values of the components of $\mathbf{e}$ (x,y,z,t) in terms of a,b,c,d. Here is a 2-d example of what I have tried so far, but I hope to do this in 4-dimensions eventually.
metric = ({{a, b},{c, d}});
eta = ({{1, 0},{0, -1}});
vb = ({{x, y},{z, t}});
neta = Transpose[vb].metric.vb; (* Need to set this equal to eta and solve for x, y, z, t *)
neta == eta
(* Need to do something like Solve[%, {x,t,y,z}] but I get {} if I do that *)
Thanks!
{a,b,c,d}
, hence is empty for "generic" values of the params. You can see this by explicitly allowing for such a situation:Solve[Flatten[neta - eta] == 0, {x, y, z, t}, MaxExtraConditions -> 1]
$\endgroup$metric
to{{a,b}, {b,d}}
but then the system is underdetermined. If you then dovb /. Solve[Flatten[neta - eta] == 0, {x, y, z, t}]
you get a result that still containsx
. $\endgroup$vb /. Solve[Flatten[neta - eta] == 0, {x, y, z, t}][[6]]
I get a matrix without x in. What's going on here? $\endgroup$Solve
thinks solution set has isolated zero dimensional components (points) as well as one dimensional families of solutions. I suspect these isolated ones are actually special values of the dimensional components though. $\endgroup$