A Blog Entry on Bayesian Computation by an Applied Mathematician
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1 Stiefel Manifolds
Stiefel manifolds \(\operatorname{St}_p(\mathbb{R}^n)\) is the set of all orthonormal \(p\)-frames in \(\mathbb{R}^n\), or equivalently, the set of all \(n\times p\) matrices with orthonormal columns. Formally, we define To make our presentation concrete, we focus on the minimal nontrivial case \[ \operatorname{St}(3,2):=\{X\in\mathbb{R}^{3\times2}\mid X^\top X=I_2\}. \] We immediately see \(\operatorname{St}(3,2)\simeq\operatorname{SO}(3)\), as \[ \operatorname{St}(3,2)\ni(x_1,x_2)\mapsto(x_1,x_2,x_1\times x_2)\in\operatorname{SO}(3), \] \[ \operatorname{SO}(3)\ni(x_1,x_2,x_3)\mapsto(x_1,x_2)\in\operatorname{St}(3,2), \] are diffeomorphisms between the two manifolds. They are 3-dimensional manifolds.
1.1 Tangent Space
Notice \(\operatorname{St}(3,2)=F^{-1}(0)\) for a analytic function \[ F(R)=R^\top R-I_2. \] For this function, the derivative is \[ DF_R(\Delta)=R^\top\Delta+\Delta^\top R. \] We see the tangent space can be expressed as \[ T_R(\operatorname{St}(3,2))=\{\Delta\in\mathbb{R}^{3\times2}\mid R^\top\Delta+\Delta^\top R=0\}. \]
1.2 Application in Control
Geodesics in \(\operatorname{SO}(3)\) is used to model dynamics of a rigid body.