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?

My problem comes from the following:

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