# A positive definite Hermitian Form

#### Try hard

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, its 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, Ive 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

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#### fresh_42

Mentor
2018 Award
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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