How to rotate a page by maintaining the same vertical orientation of the page

Question

I am looking how to rotate a page in my document. I already used landscape, but unfortunately it rotates also the pdf page. I just would like to see the text rotate with a vertical pdf page. Is there someone that could help me?

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Edit: For instance, this is the code that I am developing and the pdf plot. As you can see the pdf page is rotate and the text with the matrix is horizontal. I would like to have the page pdf vertical and the text horizontal. The reasons are because I prefer like this and I do not want to scale the matrices. Is it possible to keep the pdf page vertical and rotate the text? I already tried rotate but it does not work.

Best regards, and sorry if I am not being enough precise on it.

Current Code

\documentclass[smallextended]{svjour3}       % onecolumn (second format)
\usepackage[paperwidth=165mm, paperheight=238mm, left=2.5cm,right=2cm,top=2cm,bottom=2cm]{geometry}

\smartqed  % flush right qed marks, e.g. at end of proof
%
\usepackage[graphicx]{realboxes}
\usepackage{graphicx}
\usepackage{epstopdf}
\usepackage{lineno}
\linenumbers
\usepackage{amsmath}

\journalname{Energy Efficiency}
\usepackage{amssymb}
\usepackage[justification=centering]{caption}

\usepackage{pdflscape}
\usepackage{nomencl}
\makenomenclature


\begin{document}
    \begin{landscape}

    \section{APPENDIX}
    \label{sec:5}
    Here below the matrices

    \newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%

    \[\Scale[0.85]{
    $$
    \begin{bmatrix}
        0   \\
        0   \\
        0   \\
        ... \\
        0   \\
        0
    \end{bmatrix}^{v+1}
    \quad
    =
    \begin{bmatrix}
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        ... & ... & ... & ... & ... & ... & ... & ... & ... \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0
    \end{bmatrix}^{inv}
    \quad
    \begin{bmatrix}
        0   \\
        0   \\
        0   \\
        ... \\
        0   \\
        0
    \end{bmatrix}^{v}
    \quad
    +
    \begin{bmatrix}
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        ... & ... & ... & ... & ... & ... & ... & ... & ... \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
        0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0
    \end{bmatrix}^{inv}
    \quad
    \begin{bmatrix}
        0   \\
        0   \\
        0   \\
        ... \\
        0   \\
        0
    \end{bmatrix}^{v}
    $$
    }
    \]
    \end{landscape}
\end{document}

Current Output

Answer 1

This answer makes use of the graphicx package and its \rotatebox command. The MWE below also makes use of the more common \documentclass{article} for a wider audience with TeX.SE users.

N.B. This is untested with the \documentclass{svjour3} and other code from the MWE in your question. But I do not see any reason as to why it should fail.

Rotated Matrix (MWE)

\documentclass[a4paper]{article}
\usepackage{amsmath, amssymb}

\usepackage[justification=centering]{caption}
\usepackage{graphicx}   

\begin{document}

    \section{APPENDIX}
    \label{sec:5}
    \par Here below the matrices.

    \vspace{1em}
    \par\noindent\begin{minipage}{\linewidth}
        \centering
        \rotatebox{-90}{
            \begin{minipage}{0.8\textheight}
                \vspace*{\fill}
                \footnotesize
                \[
                \begin{bmatrix}
                    0   \\
                    0   \\
                    0   \\
                    ... \\
                    0   \\
                    0
                \end{bmatrix}^{v+1}
                \quad
                =
                \begin{bmatrix}
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    ... & ... & ... & ... & ... & ... & ... & ... & ... \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0
                \end{bmatrix}^{inv}
                \quad
                \begin{bmatrix}
                    0   \\
                    0   \\
                    0   \\
                    ... \\
                    0   \\
                    0
                \end{bmatrix}^{v}
                \quad
                +
                \begin{bmatrix}
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    ... & ... & ... & ... & ... & ... & ... & ... & ... \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   \\
                    0   & 0   & 0   & 0   & 0   & 0   & 0   & 0   & 0
                \end{bmatrix}^{inv}
                \quad
                \begin{bmatrix}
                    0   \\
                    0   \\
                    0   \\
                    ... \\
                    0   \\
                    0
                \end{bmatrix}^{v}
                \]
                \vspace*{\fill}
            \end{minipage}
        }
        \captionof{figure}{Some Matrix}
        \label{eq:somematrix}
    \end{minipage}
\end{document}

P.S. Note the useful shorthand declaration of the math packages. \usepackage{amsmath, amssymb}

Notes

• Requires \usepackage{graphicx} for the main purpose
• Requires \usepackage{caption} for the MWE's caption in the minipage environment
• \par\noindent and \vspace*{\fill} fixes horizontal alignment issues
• \footnotesize is optional. Used for demonstration only.
• Make use of TeX IDEs and their alignment features. Such as TeXStudio's table cell alignment feature (a remnant of cleaning your MWE earlier) which gives properly aligned columns in matrices and tables.
• The minipage environment in the \rotatebox command has it's height set and not it's width. So use \textheight instead.

Answer 2

To explain better the problem I post the pdf. I would like that the text would be on the bottom left corner of the page. Instead, the text is on the top side of the page. I am trying to move it down, I deleted \centering and I tried also other options I set \hspace*{3 cm} but nothing change. Could you help me please?