# How can I solve the matrix optimization problem where denominator and numerator are different?

I want to solve the following maximization problem in $$\mathbf{X}\in {\mathbb{R}}^{{m} \times {n}}$$

$$\begin{eqnarray} \begin{split} \quad\max_\mathbf{X} \frac{\mathbf{Tr}(\mathbf{X}^\top \mathbf{Q} \mathbf{X})}{\mathbf{Tr}((\mathbf{X-A})^\top \mathbf{P} (\mathbf{X-A}))} \end{split} \end{eqnarray}$$

where $$\mathbf{A}\in {\mathbb{R}}^{{m} \times {n}}$$, $$\mathbf{P}\in {\mathbb{R}}^{{m} \times {m}}$$ and $$\mathbf{Q}\in {\mathbb{R}}^{{m} \times {m}}$$ and $$\mathbf{P}$$, $$\mathbf{Q}$$ are positive definite matrix.

It is well known that if $$\mathbf{A}=\mathbf{O}$$, it is easily solved by eigenvalue problem. However, I am not sure how to solve such a problem.

• I am not sure if you require definiteness of the matrices $P$ and $Q$ for the maximum to exist. – nicoguaro Jul 23 at 22:03
• Wouldn't in the limit of $X$ to $A$ yield positive infinity for the cost function if $\text{Tr}(A^\top Q\,A)>0$, thus a maximum? Also I believe that it should be that $X,A\in\mathbb{R}^{n\times m}$ (so switching $m$ and $n$)? – fibonatic Jul 24 at 3:50