Optimization of square matrix multiplied with another matrix to have the final result a unitary matrix

Question

I have a square matrix $D$ whose size is $m \times m$ multiplied with another $m \times m$ square matrix $C$, I need to optimize the matrix $C$ to have a unitary matrix $DC$. I mean optimize the matrix $C$ such as $DC$ is a unitary matrix.

In my opinion, that can be formulated as below:

\begin{align} \min_C \|DC - Y\|^2_F&& \text{s.t.}&&(DC)(DC)^H = I_m \end{align}

where $\|\cdot\|_F$ is the Frobenius norm operator and $Y$ is any unitary matrix.

So, I don't know if we can deal with the above equality as a variant of the Procrustes problem or that's not possible. Is it possible to optimize that above equation based on $C$ following my way ? or is there another way we can set the matrix $C$ to have $(DC)(DC)^H = I_m$ ?

NB: All the matrices are real and $det(D) = 0$.

Answer 1

Could it be that you are indeed looking for the closest orthogonal matrix $Y$? Then, there is a solution which involves computing the square root of $ D^TD$ . If $E=(D^TD)^{1/2}$ were invertible, the solution would be its inverse. Yet, it is not invertible here. Then, there is a trick. If I remember well, you have to perform an eigenvalue/eigenvector decomposition of $E$, replace the null eigenvalues by $1$, you then get a novel matrix $E^*$ which is invertible, and its inverse is the (unique) solution.

If what I wrote is correct enough, I might come back with details. Meanwhile, you can look at:

• Finding the Nearest Orthonormal Matrix
• The Nearest Orthogonal or Unitary Matrix

Answer 2

If D is not singular, i.e. $\operatorname*{det}(D) \ne 0$ than simply $C = D^{-1}$ will do the trick, since $D \cdot C = I$, which is obviously unitary.

If D is singular, i.e. $\operatorname*{det}(D) = 0$ than the product is also singular, i.e. $\operatorname*{det}(D \cdot C)=0$ which means the product cannot be unitary for all possible matrices $C$.