Multiplying quadratic form using block matrix multiplication

Question

My textbook gives the following example:

The second-order cone is the norm cone for the Euclidean norm, i.e.,

$$\begin{align} C &= \{ (x, t) \in \mathbb{R}^{n + 1} \mid \left\lVert x \right\rVert_2 \le t \} \\ &= \left\{ \begin{bmatrix} x \\ t \end{bmatrix} \;\middle|\; \begin{bmatrix} x \\ t \end{bmatrix}^T \begin{bmatrix} I & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ t \end{bmatrix} \le 0, t \ge 0 \right\} \end{align}$$

The second-order cone is also known by several other names. It is called the quadratic cone, since it is defined by a quadratic inequality. It is also called the Lorentz cone or ice-cream cone.

I'm wondering how to multiply out $$\begin{bmatrix} x \\ t \end{bmatrix}^T \begin{bmatrix} I & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ t \end{bmatrix}$$

It seems to me that I need to use block matrix multiplication (assuming $I$ is the identity matrix), but even after reviewing examples (see this and Wikipedia), I keep getting the matrix $\begin{bmatrix} x^2 I - t^2 \end{bmatrix}$ as my final result, which doesn't make sense, since $x^2 I$ is an invalid matrix multiplication.

I would greatly appreciate it if people could please take the time to demonstrate this.

Answer 1

$\newcommand{\R}{\mathbb{R}}$Let $ \mathbf{0}\in\R^n$ be a column of all $0$'s, so $\mathbf{0}^T$ denotes a row of $n$ zeros. Note that by block matrix multiplication, we have $$\begin{bmatrix} I & \mathbf{0} \\ \mathbf{0}^T & -1\end{bmatrix}\begin{bmatrix} \mathbf{x} \\ t\end{bmatrix} = \begin{bmatrix}I\mathbf{x} + \mathbf{0} t\\ \mathbf{0}^T \mathbf{x} +(-1)t\end{bmatrix} = \begin{bmatrix}\mathbf{x} \\ -t\end{bmatrix}\in \R^{n+1}.$$ Also, $\begin{bmatrix}\mathbf{x} \\ t\end{bmatrix}^T = \begin{bmatrix}\mathbf{x}^T & t\end{bmatrix}\in\R^{1\times (n+1)}$ and so

$$\color{blue}{\begin{bmatrix}\mathbf{x} \\ t\end{bmatrix}^T \begin{bmatrix}\mathbf{x} \\ -t\end{bmatrix} = \begin{bmatrix}\mathbf{x}^T & t\end{bmatrix} \begin{bmatrix}\mathbf{x} \\ -t\end{bmatrix} = \mathbf{x}^T \mathbf{x} - t^2}.$$

(It looks like your error was maybe in the above step.)

Combining these facts tells us that $$\boxed{\begin{bmatrix}\mathbf{x} \\ t\end{bmatrix}^T \begin{bmatrix} I & \mathbf{0} \\ \mathbf{0}^T & -1\end{bmatrix}\begin{bmatrix} \mathbf{x} \\ t\end{bmatrix} = \mathbf{x}^T \mathbf{x} - t^2}.$$

Note that this means the inequality $$\begin{bmatrix}\mathbf{x} \\ t\end{bmatrix}^T \begin{bmatrix} I & \mathbf{0} \\ \mathbf{0}^T & -1\end{bmatrix}\begin{bmatrix} \mathbf{x} \\ t\end{bmatrix} \le 0$$ becomes $$\mathbf{x}^T \mathbf{x} - t^2\le 0,$$ which is equivalent to $\left\|\mathbf{x}\right\|_2 \le t$ (for $t\ge 0$), as was claimed.