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Every $Q\in O(3)$ with $\det Q=1$ is a rotation in three dimensions, hence is the product of two reflections in planes through its axis whose angle is half the rotation angle.

If $\det Q=-1$, its real eigenvalue is $-1$: the other two eigenvalues are either a complex-conjugate pair or two real signs, and their product is positive. Let $v$ be a corresponding unit eigenvector and let $H$ be reflection in $v^{\perp }$. Then $HQ$ fixes $v$ and has determinant $1$, so it is a rotation and hence a product of two reflections. Since $Q=H(HQ)$, every member of $O(3)$ is a product of at most three reflections.

Two reflections have determinant $1$, whereas one reflection has eigenvalues $(-1,1,1)$. The map $-I\in O(3)$ has determinant $-1$ but is not a reflection, so it cannot be a product of at most two reflections. Thus three are sometimes necessary.

The unit sphere and the kernel $e^{-c|y-x{|}^2}$ are invariant under every rotation in three dimensions fixing $y$. Therefore $T(y)$ has the same axisymmetry, so part (a) gives

Choose spherical polar coordinates with polar axis $y$ and put $u=\cos \theta =x\cdot y$. Since $|x|=|y|=1$,The tensor contraction giving the trace isSimilarly,At $c=0$ these expressions have the continuous limiting values $4\pi$ and $4\pi /3$. Writing them as $A=T_{kk}=3\alpha +\beta$ and $B=T_{ij}{y}_i{y}_j=\alpha +\beta$ gives

The orthogonal group isThe special orthogonal group is its determinant-one subgroup

Every eigenvalue of an orthogonal matrix has modulus one. A real three-by-three matrix has at least one real eigenvalue, and nonreal eigenvalues occur in a complex conjugate pair. If $Q\in SO(3)$ has such a pair $e^{\pm i\theta }$, their product is one, so the remaining eigenvalue is $\det Q=1$. If all three eigenvalues are real, each is $\pm 1$ and their product is one; an odd number of three signs with product one must include $+1$. Hence every element of $SO(3)$ has an eigenvector of eigenvalue one and represents a rotation in three dimensions about its span.

It is false that every element of $O(3)$ is either a rotation or a plane reflection. For example,is an improper orthogonal transformation combining a rotation with a reflection. Its eigenvalues are $e^{\pm i\theta }$ and $-1$, whereas a plane reflection has eigenvalues $1,1,-1$ and a rotation has determinant one.

The SO(3) group consists of real three-by-three orthogonal matrices with determinant one. It acts by rotation in three dimensions. Its group manifold is ${\mathbb{RP}}^3$, since the Adjoint double cover from SU(2) to SO(3) identifies antipodal points of the three-sphere.