Linear transformations on matrices

Abstract:

The real orthogonal group $O(n)$, the unitary group $U(n)$ and the symplectic group ${\text {sp(}}n{\text {)}}$ are embedded in a standard way in the real vector space of $n \times n$ real, complex and quaternionic matrices, respectively. Let $F$ be a nonsingular real linear transformation of the ambient space of matrices such that $F(G) \subset G$ where $G$ is one of the groups mentioned above. Then we show that either $F(x) = a\sigma (x)b$ or $F(x) = a\sigma ({x^\ast })b$ where $a,b \in G$ are fixed, ${x^\ast }$ is the transpose conjugate of the matrix $x$ and $\sigma$ is an automorphism of reals, complexes and quaternions, respectively.