A small memo about Kroenecker products and linear indexing

kroenecker_trickery.tex

\documentclass{article}
\usepackage{amsmath, amsfonts, amssymb}
\title{Kroenecker Products and Linalg Trickery}
\author{Andrea Pasqualini}

\begin{document}

\maketitle

Consider two vectors, $x$ and $y$.
\begin{align*}
  \underset{[n \times 1]}{x} && \underset{[m \times 1]}{y}
\end{align*}
Let $i$ index the elements of $x$ and $j$ index the elements of $y$.
Compute the Kroenecker product $z\equiv x \otimes y$.
The product $x_i \cdot y_j$ will be the element $z_k$, with \textcolor{red}{$k = (i - 1) \cdot m + j$}.
Explicitly, we will have
\begin{align*}
  z &\equiv \underset{[(n \cdot m) \times (1 \cdot 1)]}{x \otimes y}
  =
  \begin{bmatrix}
    x_1 \cdot y \\
    \hrulefill \\
    x_2 \cdot y \\
    \hrulefill \\
    \vdots \\
    \hrulefill \\
    x_i \cdot y \\
    \hrulefill \\
    \vdots \\
    \hrulefill \\
    x_n \cdot y
  \end{bmatrix}
  =
  \begin{bmatrix}
    x_1 \cdot y_1 \\
    x_1 \cdot y_2 \\
    \vdots \\
    x_1 \cdot y_m \\
    \hrulefill \\
    x_2 \cdot y_1 \\
    x_2 \cdot y_2 \\
    \vdots \\
    x_2 \cdot y_m \\
    \hrulefill \\
    \vdots \\
    \hrulefill \\
    x_i \cdot y_1 \\
    x_i \cdot y_2 \\
    \vdots \\
    x_i \cdot y_m \\
    \hrulefill \\
    \vdots \\
    \hrulefill \\
    x_n \cdot y_1 \\
    x_n \cdot y_2 \\
    \vdots \\
    x_n \cdot y_m
  \end{bmatrix}
\end{align*}

\end{document}