Regarding the properties of $Q$, consider this. Let $\Lambda_x = \mathrm{diag}(1_n^T X)$ and $\Lambda_y = \mathrm{diag}(1_n^T Y)$be the diagonal matrices formed with the row sum of $X$ and $Y$, respectively. Then, from ii) and iii), one can get $C 1_m = \Lambda_x 1_m$ and $1_m^TC = 1_m^T\Lambda_y$, for $C=Q^TX^TY^TQ$. These equations imply that the vector $1_m$ is a right eigenvector of $C$ with respect to $\Lambda_x$ and a left eigenvector with respecto to $\Lambda_y$.

BTW, could not figure out the typo.

Why is the problem interesting? Its solution will have relevance in land cover change detection. Each row of $X$ and $Y$ represents membership vectors of $m$ land cover types. If $X,Y\in\{1,0\}^{n\times m}$, i.e., are binary matrices with exactly one non-zero element per row, then $C=X^TY$ quantifies the changes among land cover types, and $C$ satisfies all three conditions above ($Q=I$). The problem arises when $X,Y\in[1,0]^{n\times m}$, that is partial memberships are allowed.

I understand your point. However, for a propper solution of the problem one must be able to discern between diagonal and non-diagonal matrices. So, let's just discard the scalar case. – silvanmx 0 secs ago

Looks like your answer is right! Thanks.

You are right! That solution gives actually a specific one for a given $X$. It seems Spenser's reasoning above answers the question.

Then what would be a non-diagonal 1-by-1 matrix? When I say ambiguous I mean that a scalar can be considered either diagonal or non-diagonal.

If $Y\ne X$, then $X^TWX = D(X^TX)^{-1}X^TY$ which is (I'm guessing) non-diagonal.

I meant your second interpretation. Sorry for the confusion.

Ok, I am assuming n>>m. Besides, the concept of a diagonal matrix is ambiguous for scalars.