Matrix in vector notation?

Question

Let, $\mathbf{A} \in \mathbb{R}^{n \times n}$, $x \in \mathbb{R}^{n}$, and $\mathbf{I}$ be an $n$ by $n$ identity matrix. What does,

$$ \begin{bmatrix}\mathbf{A} \\ \mathbf{I} \end{bmatrix}x, $$

mean? I see this notation often used in books and no idea what it implies. Thanks!

Answer 1

It denotes a $2n\times 1$ vector with coordinates $$ [A_1x, \ldots, A_nx, x_1,\ldots,x_n]^{T} $$ Where $A_i$ denotes the $i$-th row of matrix $A$.

Answer 2

I understand it as "expanding" the matrices $\mathbf{A}$ and the identity inside the brackets, and then, performing the matrix multiplication:

$$ \begin{bmatrix}\mathbf{A} \\ \mathbf{I} \end{bmatrix}x \equiv \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \dots \\ a_{n1} & a_{n2} & \dots & a_{nn} \\ 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \dots\\ 0 & 0 & \dots & 1 \end{bmatrix}\cdot\begin{bmatrix} x_1 \\ x_2 \\ \dots \\ x_n \end{bmatrix} $$

Technically it gives the same result as the other answer, but I find it easier to visualize it...