Minimum-Norm and Least-Squares Solutions of the Singular Sylvester Equation in Quaternions

Abstract

The quaternionic Sylvester equation $ax-xb=c$ is singular exactly when $a$ and $b$ are similar. In that case the homogeneous equation has a two-dimensional solution space, and the inhomogeneous equation is solvable only for $c$ in a two-dimensional subspace. Tian (2004) gave the general solution in this case but left open which solution has minimal norm and what the least-squares solutions are when the equation is not solvable. We answer both questions by treating $S(x)=ax-xb$ as a real-linear function on the four-dimensional Euclidean space of quaternions and working entirely within the quaternion algebra, so that every object is obtained as an explicit quaternionic expression. The solution space of the homogeneous equation is shown to be the complex line through any nonzero solution $x_1$, with the orthogonal basis $x_1$ and $x_1(\mathrm{Im}b)$ in every case, including $b=a^{\ast }$, which the known two-parameter formula excludes. In the singular case $S$ is skew-symmetric and satisfies $S^3=-4|\mathrm{Im}a{|}^{2}S$. From these two facts we obtain the solvability of $ax-xb=c$ as a Fredholm alternative, namely $c$ orthogonal to the kernel of $S$, which is equivalent to Tian's condition $ac=c{b}^{\ast }$. The Moore-Penrose inverse is $S^{+}=-S/(4|\mathrm{Im}a{|}^2)$, and the orthogonal projectors onto the range and the kernel of $S$ are given in closed form. This yields the minimum-norm solution and all least-squares solutions for every right-hand side $c$, the former being the first term of Tian's general solution. Finally, the second basis vector $x_1(\mathrm{Im}b)$ is a real multiple of the quaternion square root of $(\mathrm{Im}a)(\mathrm{Im}b{)}^{\ast }$, which after normalisation is the unit quaternion rotating $\mathrm{Im}b$ onto $\mathrm{Im}a$ along the shortest arc.