Could you please help me to understand how the second part of the equation (quadratic form) derived form the first one?
The basic definition of the second-order cone is:
$C = \big\{(x,t) \in \mathbb{R}^{n+1} | \|x\|_2 \leq t \big\}= \big\{(x,t) \in \mathbb{R}^{n+1} | x^Tx \leq t^2 \big\}= \Bigg\{ \begin{pmatrix} x\\ t \end{pmatrix} \bigg| \begin{pmatrix} x\\ t \end{pmatrix}^T \begin{pmatrix} I & 0\\ 0 & -1 \end{pmatrix}\begin{pmatrix} x\\ t \end{pmatrix}\leq 0, t \geq 0\Bigg\}$
Hint: Why don't you try expanding that quadratic form in terms of $x$ and $t$. \begin{align} \begin{pmatrix} x\\ t \end{pmatrix}^T \begin{pmatrix} I & 0\\ 0 & -1 \end{pmatrix}\begin{pmatrix} x\\ t \end{pmatrix} \end{align} First expand \begin{align} \begin{pmatrix} I_{n\times n} & 0_{n\times 1}\\ 0_{1\times n} & -1 \end{pmatrix} \begin{pmatrix} x_{n\times 1}\\ t \end{pmatrix} \end{align} using the rules of block matrix multipication and do again the same using the term on the left side, then note that $x^Tx=||x||_2^2$.