<s>[INST] Second order cone with quadratic interpretation 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:
$C = \big\{(x,t) \in \mathbb{R}^{n+1} | \|x\|_2 \leq t \big\}= \big\{(x,t) \in \mathbb{R}^{n+1} | x^Tx \leq t^2 \big\}= \Bigg\{
\begin{pmatrix}
    x\\
    t
\end{pmatrix}
\bigg|
\begin{pmatrix}
    x\\
    t
\end{pmatrix}^T
\begin{pmatrix}
    I & 0\\
    0 & -1
\end{pmatrix}\begin{pmatrix}
    x\\
    t
\end{pmatrix}\leq 0,
t \geq 0\Bigg\}$ 
 [/INST] 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 $x^Tx=||x||_2^2$. 
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