= Axis decomposition of a symmetric matrix
{title2=$T=AI+Bvv^T+Cv^T+vC^T+D$}
Fix $v\ne0$, put $s=v^Tv$, $t=v^TTv$, and $P=I-vv^T/s$. Any real <symmetric matrix> on three-dimensional <Euclidean space> has a unique decomposition with $C^Tv=0$, $Dv=0$ and $\operatorname{tr}D=0$:
$$
A=\frac12\left(\operatorname{tr}T-\frac ts\right),\qquad B=\frac{t/s-A}{s},\qquad C=\frac{PTv}{s},\qquad D=PTP-AP.
$$
The transverse restriction is a symmetric two-dimensional matrix, whose <traceless matrix> part has dimension two. Together the two scalars, two components of $C$ and two components of $D$ give the six parameters of $T$.
Back to article page
