[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.1

For fixed basis of in $scrH$ and $scrK$, the matrix $A^*$ is the conjugate transpose of the matrix of $A$.

 

Solution.  $$eex ea (A^*)_{ij}&=e_i^*A^*f_j\ &=sef{e_i,A^*f_j}_scrH\ &=sef{Ae_i,f_j}_scrK\ &=overline{sef{f_j,Ae_i}}\ &=overline{a}_{ji}. eea eeex$$