Final answer:
The symplectic group Sp(n) is not compact. Instead, the invariance of the distance of point P to the origin under rotations can be shown by using the properties of orthogonal matrices in the special orthogonal group SO(n).
Step-by-step explanation:
The symplectic group Sp(n) is actually not compact for any integer n > 0. The question contains a typo or misunderstanding. Instead, one can show that the special orthogonal group SO(n), which is the group of n × n orthogonal matrices with determinant 1, is compact. To demonstrate that the distance of a point P to the origin is invariant under rotations, let's consider a point P in the coordinate system with coordinates (x, y, z). The distance squared from P to the origin is x² + y² + z².
Under a rotation represented by an orthogonal matrix, the coordinates of P change, but the quadratic form remains the same due to the properties of orthogonal matrices. Specifically, for an orthogonal matrix Q, we have Q·Qᵠ = I, where I is the identity matrix, and Qᵠ is the transpose of Q. For any vector v, applying the rotation Q to v gives us Qv, and the squared length of Qv is given by (Qv)·(Qv)ᵠ = v·Qᵠ·Q·v = v·I·v = v·v, which is the original squared length. Hence, the distance remains invariant under rotation.