Suppose   

 - $\{(x_1,x_2) : x_1^2+x_2^2 = 1\}$   the unit circle. 


 
 
Consider two sets defined by a quadratic constraint and LMI:  
 

 - $$\{Y\in R^{2\times 2}: \begin{bmatrix}x_1 & x_2 \end{bmatrix}\begin{bmatrix}Y_{11}+Y_{22} & Y_{21}-Y_{12} \\ Y_{21}-Y_{12}&-Y_{11}-Y_{22} \end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \end{bmatrix}\leq 1\}$$   
 -  $$\{Y\in R^{2\times 2}: \begin{bmatrix}Y_{11}+Y_{22} & Y_{21}-Y_{12} \\ Y_{21}-Y_{12}&-Y_{11}-Y_{22} \end{bmatrix}\preceq I\}$$  

>How to show both constraints define the same set?   
 
Suppose $$\begin{bmatrix}Y_{11}+Y_{22} & Y_{21}-Y_{12} \\ Y_{21}-Y_{12}&-Y_{11}-Y_{22} \end{bmatrix}=I,$$ then the equality of the first constraint holds.  

Moreover, let $Y_{11}+Y_{22}=a,  Y_{21}-Y_{12}=b$, we can rewrite the first constraint as:   

$$(x_1^2-x_2^2)a+2x_1x_2b\leq 1$$  How to do the next step?


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My problem comes from the following:  

http://arxiv.org/abs/1403.4914 (p.1328 top)