Denote by Hn the linear space of n×n self-adjoint complex matrices, and by Pn the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on Hn
\langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n)
and its derived metric. Show that every ϕ:Pn→Pn isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as
\phi(A) \equal{} UAU^* \plus{} X\qquad (A\in H_n)
or
\phi(A) \equal{} UA^TU^* \plus{} X\qquad (A\in H_n)
where U is an n×n unitary matrix, X is a positive-semidefinite matrix, and T and ∗ denote taking the transpose and the adjoint, respectively. geometry3D geometrylinear algebramatrixlinear algebra unsolved