Solution of the problem of optimal diagonal scaling for quasireal Hermitian positive-definite $3\times3$ matrices

The paper completely solves the problem of optimal diagonal scaling for quasireal Hermitian positive-definite matrices of order 3. In particular, in the most interesting irreducible case, it is demonstrated that for any matrix $C$ from the class considered there is a uniquely determined optimally scaled matrix $D^*_0CD_0$ of one of the four canonical types, and formulas for the entries of the diagonal matrix $D_0$ are presented as well as formulas for the eigenvalues and eigenvectors of $D^*_0CD_0$ and for the optimal condition number of $C$, which is equal to $k(D^*_0CD_0)$. The optimality of the Jacobi scaling is analyzed.