Let
B=(acbd)∈U⊥
Then B is orthogonal to any element in U and, in particular, to the basis elements. Thus,
Tr((acbd)⋅(100−1))=Tr(ac−b−d)=a−d=0
Thus, a=d . Similarly, dotting with the other basis elements, we get the two following equations:
Tr((acbd)⋅(0010))=Tr(00ac)=c=0
Tr((acbd)⋅(0100))=Tr(bd00)=b=0
Thus, using the fact that a=d and c=b=0 , we get that every matrix in U⊥ is of the form
(a00a)
Thus, U⊥ is spanned by the identity matrix.
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