MathDB

Denote by

H_n

the linear space of

n\times n

self-adjoint complex matrices, and by

P_n

the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on

H_n

\langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n) and its derived metric. Show that every

\phi: P_n\to P_n

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\times n

unitary matrix,

X

is a positive-semidefinite matrix, and

^T

and

^*

denote taking the transpose and the adjoint, respectively.

Problem Statement