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The context for the question is in the attachments (pg1.png, pg2.png, pg3.png), so there is some reading involved. Although, it is a short and simple read if anything. The inquiry is in (inquiry.png).
My understanding of the situation is that Q(t) abides by the differential equation
Q'(t)Q(t)^{T} + Q(t)Q'(t)^{T} = 0 .
So by something akin to Picard's existence theorem on differential equations, we can resolve exact trajectories for Q(t) given initial conditions. The peculiar thing is that
Q(t_{0}) z(t_{0}) = z(t_{0})
can likely be assumed to be true for all z(t_{0}) ∈ ℝ^{n}, in which case
Q(t_{0}) = I
should also be true.
Now if I let R = ( i(t_{0}) , j(t_{0}) , k(t_{0}) ), then
R^{T}R = RR^{T} = I ,
and the question insists that
R^{T}Q(t)R = Q ⇒ Q(t)R = RQ(t).
Although, our choice of the initial condition basis vectors for R is arbitrary, so how can we expect that R necessarily commutes with Q(t)? Unless of course, the question is asking for the matrix expression of Q(t) in the { i(t_{0}) , j(t_{0}) , k(t_{0}) } basis. I am just wondering if I am to expect the matrix expression of Q(t) to remain invariant given any orthonormal basis of the same orientation?
(Also, how do I insert latex?)
My understanding of the situation is that Q(t) abides by the differential equation
Q'(t)Q(t)^{T} + Q(t)Q'(t)^{T} = 0 .
So by something akin to Picard's existence theorem on differential equations, we can resolve exact trajectories for Q(t) given initial conditions. The peculiar thing is that
Q(t_{0}) z(t_{0}) = z(t_{0})
can likely be assumed to be true for all z(t_{0}) ∈ ℝ^{n}, in which case
Q(t_{0}) = I
should also be true.
Now if I let R = ( i(t_{0}) , j(t_{0}) , k(t_{0}) ), then
R^{T}R = RR^{T} = I ,
and the question insists that
R^{T}Q(t)R = Q ⇒ Q(t)R = RQ(t).
Although, our choice of the initial condition basis vectors for R is arbitrary, so how can we expect that R necessarily commutes with Q(t)? Unless of course, the question is asking for the matrix expression of Q(t) in the { i(t_{0}) , j(t_{0}) , k(t_{0}) } basis. I am just wondering if I am to expect the matrix expression of Q(t) to remain invariant given any orthonormal basis of the same orientation?
(Also, how do I insert latex?)
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