It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $ A \mapsto PAQ+R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$. The result extends previously known theorems where the map was assumed to be also injective.

Let $U(n)$ be the group of $n\times n$ unitary matrices. We show that if $\phi$ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is a multiple of $n$, and $\phi$ has the form $$ A \mapsto V[(A\otimes I_s)\oplus (A^t \otimes I_{r})]W $$ for some $V, W \in U(m)$. From this result, one easily deduces the characterization of linear operators that map $U(n)$ into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$. The $c$-numerical range of an $n \times n$ matrix $A$ is defined by $$ W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\} \text{ an orthonormal basis for } \IC^n \Bigr\}, $$ and the $c$-numerical radius of $A$ is defined by $r_c (A) = \max \{|z| : z \in W_c (A)\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying $$ W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or} \quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A. $$