y\_4, y\_4, y\_4, y\_4, y\_5)  ,\
T\^&=&[diag]{}([**X\_4**]{}, [**X\_5**]{}, [**X\_5**]{}, y\_2, y\_3, y\_3, y\_4, y\_4)  ,\
T\^&=&[diag]{}( y\_3, y\_2,y\_2)  ,\
T\^&=&2 y\_2  . Here ${\bf X_i}$ are the $2 \times 2$ matrices whose eigenvalues $x_i^\pm$ are given by x\_1\^&=& 12\_1+22\_2+14\_4 , \[x1\]\
x\_2\^&=& 4\_1+4\_2-2\_4 ,\
x\_3\^&=& 4\_2+4\_1 ,\
x\_4\^&=& 8\_1+4\_2-2\_4 ,\
x\_5\^&=& 12\_2+14\_42 . The eigenvalues $y_i$ are obtained as y\_1&=&8\_2+16\_4 ,\
y\_2&=&8\_2+4\_4 ,\
\[b2\] y\_3&=&4\_3+\_5 ,\
y\_4&=&4\_3-2\_5  ,\
y\_5&=&4(\_3+\_5) ,\
y\_6&=& 8\_2+4( 2+ )\_4  ,\
y\_7&=& 8\_2+4( 2- )\_4  . \[y5\] Although the transition matrix between initial and final two-body states is originally $91 \times 91$, the number of independent eigenvalues turns out to be only seventeen.

Unitarity bounds on the masses
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In this section we analyze mass bounds on the Higgs bosons in the GM model, imposing the condition of perturbative unitarity in Eq. (\[PU\]) to the transition matrix given in the previous section. Consequently we obtain seventeen inequations with respect to all the independent eigenvalues of the transition matrix $T$ in Eq.(\[Tmatrix\]) as |x\_1\^|, |x\_2\^|, |x\_3\^|, |x\_4\^|, |x\_5\^|, |y\_1|, |y\_2|, |y\_3|, |y\_4|, |y\_5|, |y\_6|, |y\_7| &lt; 8. \[PUconstraint\] These eigenvalues are respectively given in Eqs. (\[x1\]) - (\[y5\]) as a combination of the dimensionless coupling constants $\lambda_i$ 