Denote U a 3-dimensional Unitary Matrix over the complex numbers. Denote D1 and D2 as two 3-dimensional matrixes that are diagonal and whose entries are of complex norm one. Denote O1 and O2 as two 3-dimensional orthogonal matrices i.e. real entries only. Denote X as the diagonal matrix with diagonal entries (1, i, i) where i is the imaginary unit.
Given any U, can it always written in the form U = D1 O1 X O2 D2?
There are no assumed relationships between the entries of D1, O1, O2, and D2 they can be totally unrelated numbers. I.e. O2 need not have any relation to O1 or its transpose or inverse or angles, etc.. Nor need the entries of D2 in anyway relate to those of D1.
Let Y = the diagonal matrix with diagonal matrix entries (1,1,z). Given U, I can always write it in the form D1 O1 Y O2 D2 where z has complex norm one; but I can't figure out if there is another decomposition of the desired form above; with totally different matrices D1, O1, O2, D2.