there in fact isn’t any matrix X that could reasonably be considered a D3/2, because such an X should satisfy X2=D3, but the matrix D3 does not have a square root (see e.g. https://math.stackexchange.com/a/66156/540174 for how to think about this)
I think this approach doesn’t make sense. Issues, briefly:
if you want to be squaring D, you need it to be square — you should append another row of 0s
this matrix D does not have a logarithm, because it isn’t invertible ( https://en.wikipedia.org/wiki/Logarithm_of_a_matrix#Existence )[1]
there in fact isn’t any matrix X that could reasonably be considered a D3/2, because such an X should satisfy X2=D3, but the matrix D3 does not have a square root (see e.g. https://math.stackexchange.com/a/66156/540174 for how to think about this)
also, note that it generally doesn’t make sense to speak of the log of a matrix — a matrix can have (infinitely) many logarithms ( https://en.wikipedia.org/wiki/Logarithm_of_a_matrix#Example:_Logarithm_of_rotations_in_the_plane )