Alterando o estilo da matriz

Question

Tenho as seguintes matrizes com seu respectivo código. A imagem anexada mostra a saída. Como tenho que colocar essa imagem em uma tese acadêmica, pensei que meu design atual está um pouco exagerado, em termos de estilo, ou seja, preciso fazer com que pareça mais "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}

Isto é o que tenho em 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}

Estou tendo problemas com os rótulos Filmes, Usuários e Fatores. Você poderia me dar uma mão nisso?

ps: Também pensei nesse, mas seria ótimo adicionar mais k fatores:

Answer 1

Usando arrayerotatebox

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

Talvez você tenha gostado:

\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}