Problem with minimizing the matrix norm

[mnb96]

Hello,

I have to to find the entries of a matrix $X\in \mathbb{R}^{n\times n}$ that minimize the functional: $Tr \{ (A-XB)(A-XB)^* \}$, where Tr denotes the trace operator, and ^* is the conjugate transpose of a matrix. The matrices A and B are complex and not necessarily square.

I tried to reformulate the problem with Einstein notation, then take the partial derivatives with respect to each $a^{i}_{j}$ and set them all to zero. The expression becomes pretty cumbersome and error-prone.

I was wondering if there is an easier and/or known solution for this problem.
Thanks.

 

[lpetrich]

In effect, minimizing Tr((A - B.X).(A - B.X)⁺) where the + means Hermitian conjugate or transpose conjugate. First one expands it:

Tr(A.A⁺) - Tr(A⁺.B.X) - Tr(X⁺.B⁺.A) + Tr(B⁺.B.X.X⁺)

Next, consider how X varies: X = X_r + i*X_i where X_r and X_i are the real and imaginary parts.

The Hermitian conjugate X⁺ = (X_r - i*X_i)^T and it is evident that one can treat X and X⁺ as separate variables, since they are different linear combinations of X_r and X_i. Differentiating by X⁺ and X yields

B⁺.B.X= B⁺.A
X⁺.B⁺.B = A⁺.B

 

[mnb96]

Thanks lpetrich,

I actually got the same result few hours before I found your reply, though you actually showed how to solve a more general problem where also X is complex.
What I did was basically to use Einstein notation to derive some useful matrix calculus identities, and then by using them it was kind of easy to arrive at the final formula(s) that you wrote.
It was interesting to see how using a different notation made the problem more manageable.