A positive definite Hermitian Form

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In this question I let "x1t , x2t, x3t " be the conjugate of x1, x2, x3

The hermitian form
Hc(x) = c*x1t*x1 + 2*x2t*x2 - i*x1t*x2 + i*x2t*x1 + x1t*x3 + x3t*x1
+i*x2t*x3 - i*x3t*x2 (sorry, it`s a bit messy)

For which value of c is Hc ositive definite?

I have tried to find the eignvalue in terms of c by trying to solve the
charactiristic polynomial, but seems too complicate to do, I`ve also tried
to solve by completing the square but not so successful.
So is there any way to solve this without using computer softwares like
maple? Thanks
 

fresh_42

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The determinant ##\operatorname{det}(H_c-\lambda I)=0## yields (if I didn't make a mistake)
\begin{align*}
0&=(\lambda-u)(\lambda-v)(\lambda-w)\\&=\lambda^3-(u+v+w)\lambda^2+(uv+uw+vw)\lambda-uvw \\
&=\lambda^3-(2+c)\lambda^2+(2c-3)\lambda +c
\end{align*}
For positive definiteness we need ##u,v,w > 0##, i.e. ##uv+uw+vw =2c-3 >0 ## and ##uvw=-c>0##, thus ##\frac{3}{2} < c < 0## which is not possible.
 

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