Improved M-matrix condition for the existence of positive-real weighting

It is proved that, for a given stable transfer matrix G(s), there exists a constant diagonal matrix W which makes WG(s) positive-real if Re g_ii(jω) ≥ 0 and I?? is an M-matrix where ? = (?_jk) is defined by ?_ii = 0 and $\text{?}{}_{\mathrm{jk}}\text{=}\text{sup}{}_{\mathrm{\omega }}\text{|g}{}_{\mathrm{ik}}\text{(gω) |/(}\text{Re}\text{[g}{}_{\mathrm{ii}}\text{(jω)]·}\text{Re}\text{[g}{}_{\mathrm{kk}}\text{(jω])}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.