Given $\mathbf{A}$ and $\mathbf{B}$ two $m \times n$ real matrices, is there a closed form for the matrix equation \begin{equation} \|\mathbf{X}\|^{2}_{F} - 2 \cdot \mbox{trace} (\mathbf{X}^T\mathbf{A}) +\|\mathbf{B}\|^{2}_{F} = 0, \end{equation}
where $\mathbf{X} \in \mathbb{R}^{m\times n}$ and $\| \cdot \|_{F}$ is the Frobenius norm? Notice that this equation can be viewed as a generalization of a simple quadratic equation.
$||X||_F^2=Tr(X^TX)$. So you can rewrite your equation in the following form: $$ Tr(X^TX-2X^TA+B^TB)=Tr(X^TX-X^TA-A^TX+A^TA+B^TB-A^TA)=Tr((X-A)^T(X-A)+B^TB-A^TA)=0 $$ So $$ Tr(X-A)^T(X-A))=Tr(A^TA-B^TB); $$ At least we can understand now that solution may exists only if $a=Tr(A^TA-B^TB)\ge 0$. Now all matrices such that $||Y||_F^2=a$ will generate solution as $X=Y+A$.