Cambiar el estilo de la matriz

Question

Tengo las siguientes matrices con su respectivo código. La imagen adjunta muestra el resultado. Como tengo que poner esa imagen en una tesis académica, pensé que mi diseño real está un poco pasado de la raya, en términos de estilo, es decir, necesito que parezca más "matemático".

\documentclass{standalone}
%\usepackage[showframe]{geometry}
\usepackage{tikz}
\definecolor{gris}{RGB}{250,250,250}
\usepackage{lmodern}
\usetikzlibrary{matrix,
    positioning}
\tikzset{
    mymatrix/.style = {matrix of nodes,
        nodes in empty cells,
        nodes={minimum height=3ex, minimum width=1em,
            inner sep=0pt, outer sep=0pt, anchor=center,
            draw, very thin,scale=0.8,transform shape},
        column sep=0pt,
        row sep=0pt,
        inner sep=0.5\pgflinewidth, outer sep=1pt,
        draw, thick},
    empty node/.style = {draw,fill=gris},
    every label/.append style = {font=\large, text=teal}
}
\makeatletter
\def\tikz@lib@matrix@empty@cell{%
    \iftikz@lib@matrix@empty%
    \node[name=\tikzmatrixname-\the\pgfmatrixcurrentrow-\the\pgfmatrixcurrentcolumn,empty node]{};\fi}
\makeatother

\usepackage{bm}
\newcommand{\MatrixVariable}[1]{\bm{{#1}}}

\begin{document}

    \begin{tikzpicture}[node distance=0pt,scale=0.6,nodes={transform
        shape}]
    \matrix (m1) [mymatrix,
    label={[rotate=90, anchor=south,black]left: Movies},
    label={[black]below:$\MatrixVariable{{R}}$},
    label={[black]above:Users},
    nodes={fill=gray!30}]
    {
        1   &   & 3 &   &   & 5 &   &   & 5 &   & 4 &   \\
        &   & 5 & 4 &   &   & 4 &   &   & 2 & 1 & 3 \\
        2   & 4 &   & 1 & 2 &   & 3 &   & 4 & 3 & 5 &   \\
        & 2 & 4 &   & 5 &   &   & 4 &   &   & 2 &   \\
        &   & 4 & 3 & 4 & 2 &   &   &   &   & 2 & 5 \\
        1   &   & 3 &   & 3 &   &   & 2 &   &   & 4 &   \\
    };
    \node (approx) [right=of m1,scale=2] {$\approx$};
    \matrix (m2) [mymatrix=none, right=0.1em of approx,
    label={[rotate=90, anchor=south east,xshift=-0.7em,black]left: Movies},
    label={[black]below:$\MatrixVariable{{Q}}$},
    label={[black]above:$f$-factors},
    nodes={minimum width=2.2em,fill=white}]
    {
        0.1& -0.4  & -0.2 \\
        -0.5& 0.6 & 0.5 \\
        -0.2& 0.3 &0.5  \\
        1.1& 2.1 & 0.3 \\
        -0.7&2.1  & -2 \\
        -1& 0.7 & 0.3 \\
    };
    \node (times) [right=of m2,scale=2] {$\cdot$};
    \matrix (m3) [mymatrix=none, right=of times,
    label={[rotate=90, anchor=north,black]right: $f$-factors},
    label={[black]below:$\MatrixVariable{{P}^{T}}$},
    label={[black]above:Users},
    nodes={minimum width=2em,fill=white}]
    {
        1.1&-2  & 0.3 & 0.5 &-2  &  -0.5 & 0.8 & -0.4 & 0.3 & 1.4 & 2.4 &  -0.9\\
        -0.8& 0.7 & 0.5 & 1.4 &0.3  & -1 & 1.4 & 2.9 & 0.7 & - 1.2 & 1.3& 0.4\\
        2.1& -0.4 & 0.6 & 1.7 & 2.4 & 0.9 & -0.3 & 0.4 &  0.8 &   0.7 & -0.6&0.1 \\
    };
    \end{tikzpicture}

\end{document}

Esto es lo que tengo en mente:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
    \[
    \begin{pmatrix}
    x_{11}       & x_{12} & x_{13} & \dots & x_{1n} \\
    x_{21}       & x_{22} & x_{23} & \dots & x_{2n} \\
    \hdotsfor{5} \\
    x_{d1}       & x_{d2} & x_{d3} & \dots & x_{dn}
    \end{pmatrix}
    =
    \begin{pmatrix}
    x_{11} & x_{12} & x_{13} & \dots  & x_{1n} \\
    x_{21} & x_{22} & x_{23} & \dots  & x_{2n} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    x_{d1} & x_{d2} & x_{d3} & \dots  & x_{dn}
    \end{pmatrix}
    \]
\end{document}

Tengo problemas con las etiquetas Películas, Usuarios y factores. ¿Podrías echarme una mano con este?

PD: También pensé en este, pero sería genial agregar más factores k:

Answer 1

Usando arrayyrotatebox

MWE

\documentclass{article}
\usepackage{amsmath}
\usepackage{graphicx}
\begin{document}
\[\begin{array}{ccc}
 \text{Users} & &  \text{factors}\\
    \rotatebox[origin=c]{90}{\text{Movies}}
    \begin{pmatrix}
    x_{11}       & x_{12} & x_{13} & \dots & x_{1n} \\
    x_{21}       & x_{22} & x_{23} & \dots & x_{2n} \\
    \hdotsfor{5} \\
    x_{d1}       & x_{d2} & x_{d3} & \dots & x_{dn}
    \end{pmatrix}
    &=&
    \rotatebox[origin=c]{90}{\text{Movies}}
    \begin{pmatrix}
    x_{11} & x_{12} & x_{13} & \dots  & x_{1n} \\
    x_{21} & x_{22} & x_{23} & \dots  & x_{2n} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    x_{d1} & x_{d2} & x_{d3} & \dots  & x_{dn}
    \end{pmatrix}
\end{array}\]
\end{document}

Answer 2

Quizás te haya gustado:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
    \[
    \underbrace{\!
    \begin{pmatrix}
    x_{11} & x_{12} & \dots  & x_{1n} \\
    x_{21} & x_{22} & \dots  & x_{2n} \\
    \vdots & \vdots & \ddots & \vdots \\
    x_{d1} & x_{d2} & \dots  & x_{dn}
    \end{pmatrix}\!
    }_{\mathbf{R}_{\text{Movies}\times \text{Users}}}
    \approx
    \underbrace{\!
    \begin{pmatrix}
    y_{11} & y_{12} & \dots  & y_{1n} \\
    y_{21} & y_{22} & \dots  & y_{2n} \\
    \vdots & \vdots & \ddots & \vdots \\
    y_{d1} & y_{d2} & \dots  & y_{dn}
    \end{pmatrix}\!
    }_{\mathbf{Q}_{\text{Movies}\times f\text{-factors}}}
    \cdot
    \underbrace{
    \begin{pmatrix}\!
    z_{11} & z_{12} & \dots  & z_{1n} \\
    z_{21} & z_{22} & \dots  & z_{2n} \\
    \vdots & \vdots & \ddots & \vdots \\
    z_{d1} & z_{d2} & \dots  & z_{dn}
    \end{pmatrix}\!
    }_{\mathbf{P}^T_{f\text{-factors}\times\text{Users}}}
    \]
\end{document}