TWO RESULTS ON $C$-CONGRUENCE NUMERICAL RADII

Let $M_n$ denote the set of $n\times n$ complex matrices. For $A$ and $C$ in $M_n$, define the $C$-congruence numerical radius of $A$ by\[ \rho_c (A) = \max\{|tr(CUAUt)|: U \text{ is unitary}\}. \]First, we show that $\rho_c$ is a norm on $M_n$ if and only if $C$ is neither symmetric nor skew-symmetric. Secondly, we use $\rho_c$ to characterize two matrices $A$ and $B$ in $M_n$ to be unitarily congruent (i.e. $A= UBU^t$ for some unitary $U$).