Assuming we know vectors $a, b$ and want to find matrix $X$ which Frobenius norm is some constant k satisfy this equation:$XX^Ta-Xb=0$? the dimension of X is $m*n$, the dimension of $a$ is $m*1$, the dimension of $b$ is $n*1$.
I know the solutions of $XX^Ta-Xb=0$ have infinite many, but the constraint of the constant norm of matrix $X$ should help to eliminate many solutions that do not have the constant norm k.