Looking at a basis and their transformed.
I have two independently evaluated matrix expressions, one matrix is formed out of pairwise dot products of a transformed basis:
nmatrix=Outer[Simplify[
TensorExpand[#2 . #1]] &, basisnTransformed, basisnTransformed]
and the matrix product of the actual transformation matrix R and its transpose:
R . Transpose[R]
both turn out to be equal and Mathematica gives me the correct answer:
nmatrix == R.Transpose[R] -> True
which is correct.
Now I want to get a step further and say
orthonormalTransformedConditions = {nmatrix == IdentityMatrix[3]}
So nmatrix or the pairwise dot products of a transformed basis is assumed to be the IdentityMatrix[3} in other words I assume the transformed basis is orthnormal and then show that R is also orthonormal by proving
R.Transpose[R] == IdentityMatrix[3]
but the test:
Simplify[
Equal[R . Transpose[R], IdentityMatrix[3],
Assumptions -> orthonormalTransformedConditions]]-> Flase
Why ?