Beamer: matrix with different properties for first row

Question

I want to have a slide that looks as follows:

• At the top, I'd like to have a theorem statement in a tcbtheorem environment I've defined.
• Below that, I'd like, after some vertical spacing, two columns of the following form.
• The first row must have each element be a tcolorbox of the form I've defined; further, each of these should be able to hold a couple of lines of text in them, while not getting too wide (so they might have somewhat larger height than a box that has only line of text).
• The subsequent rows should be tcolorboxes with a more plain look than those in the first row (I'd like to be able to customize these; so essentially, I'm asking for a different style in these rows).
• I want the entire matrix to be revealed in the order: (row1, col1), (row1, col2), (row2, col1), (row2, col2), etc.

So far, I've managed to achieve very few of these goals. I'd really appreciate any help.

\documentclass{beamer}
\mode<presentation>
{
  \setbeamertemplate{navigation symbols}{}
  \setbeamertemplate{caption}[numbered]
}

\usepackage{tcolorbox}
\usepackage{tikz}
\usetikzlibrary{matrix,overlay-beamer-styles}

%%%%%%%%%
\usetikzlibrary{calc}
%%%%%%%%
\tcbuselibrary{theorems}


\newtcbtheorem[]{mylemmma}{Lemma}{colframe=green,colback=white, width=\textwidth}{lem}
\tcbset{colframe=green, colback=white}  
\resetcounteronoverlays{tcb@cnt@mylemmma}

\begin{document}
\section{Proofs}
\newtcbtheorem[]{mydef}{Problem Statement}{colframe=white, colback=white}{ps}
\tcbset{colframe=black, colback=white}              

\resetcounteronoverlays{tcb@cnt@mydef}

\begin{frame}[fragile]
    \begin{mylemmma}{theorem title}{thm}
     For any fixed iteration, 
    \[
    f(x+a)\approx f(x) + f'(x)(a) + 1/2 f''(z) (a^2)
    \]
    \end{mylemmma}

    \centering
\begin{tikzpicture}[optimization problem/.style = {%
            rounded corners, 
            draw = blue,
            thick,
            fill= white,
            inner ysep=5pt,
            inner xsep=5pt,
            align = center},
            ]

\matrix[matrix of nodes,row sep=1em,column sep=1em,
    nodes={anchor=west,
    visible on=<\pgfmatrixcurrentrow->,},
    column 1/.style={nodes={optimization problem,
    }}, 
    column 2/.style={nodes={optimization problem,
    alt=<-\the\numexpr\pgfmatrixcurrentcolumn-1>{opacity=0,text opacity=0}{},
    }}](M){
    vectors   &  matrices\\
    reason 1    & matrix reason 1\\
    reason 2    & matrix reason 2\\
};          

\end{tikzpicture}

\end{frame}


\end{document}

Answer 1

I am a big fan of one question per question, so this answer focuses on the last question because the answer is very simple.;-) The answer is

 visible on=<\the\numexpr-2+\pgfmatrixcurrentcolumn+2*\pgfmatrixcurrentrow\relax->

(where I added a -2 thanks for your comment!) or, in more detail,

\documentclass{beamer}
\mode<presentation>
{
  \setbeamertemplate{navigation symbols}{}
  \setbeamertemplate{caption}[numbered]
}

\usepackage{tcolorbox}
\usepackage{tikz}
\usetikzlibrary{matrix,overlay-beamer-styles}

%%%%%%%%%
\usetikzlibrary{calc}
%%%%%%%%
\tcbuselibrary{theorems}


\newtcbtheorem[]{mylemmma}{Lemma}{colframe=green,colback=white, width=\textwidth}{lem}
\tcbset{colframe=green, colback=white}  
\resetcounteronoverlays{tcb@cnt@mylemmma}

\begin{document}
\section{Proofs}
\newtcbtheorem[]{mydef}{Problem Statement}{colframe=white, colback=white}{ps}
\tcbset{colframe=black, colback=white}              

\resetcounteronoverlays{tcb@cnt@mydef}

\begin{frame}[fragile]
    \begin{mylemmma}{theorem title}{thm}
     For any fixed iteration, 
    \[
    f(x+a)\approx f(x) + f'(x)(a) + 1/2 f''(z) (a^2)
    \]
    \end{mylemmma}

    \centering
\begin{tikzpicture}[optimization problem/.style = {%
            rounded corners, 
            draw = blue,
            thick,
            fill= white,
            inner ysep=5pt,
            inner xsep=5pt,
            align = center},
            ]

\matrix[matrix of nodes,row sep=1em,column sep=1em,
    nodes={anchor=west,optimization problem,
    visible on=<\the\numexpr-2+\pgfmatrixcurrentcolumn+2*\pgfmatrixcurrentrow\relax->,},
    ](M){
    vectors   &  matrices\\
    reason 1    & matrix reason 1\\
    reason 2    & matrix reason 2\\
};          

\end{tikzpicture}

\end{frame}
\end{document}

How does that work? First of all, in tikzlibrarymatrix.code.tex you can find a couple of nice keys and objects which have not made their way to the pgfmanual yet. In this answer, the important ones are

• \pgfmatrixcurrentcolumn, a TeX count, which indicates the column of the cell, and
• \pgfmatrixcurrentrow, a TeX count indicating the row of the column.

Apart from that, you can find the keys

• execute at end matrix,
• execute at begin cell,
• execute at end cell,
• execute at empty cell,

which are hooks that can be used to fill matrices automatrixally. In the above,

visible on=<\the\numexpr-2+\pgfmatrixcurrentcolumn+2*\pgfmatrixcurrentrow\relax-> 

uses the visible on key from the overlay-beamer-styles library. The combination is chosen such that for overlay number 1 the cell 11 becomes visible since -2+column+2*row=1 for column=row=1, for overlay number 2 cell 12 becomes visible since -2+column+2*row=2 for column=2 and row=1, and so on. If you have a matrix with 7 columns, say, you will need thus

 visible on=<\the\numexpr-7+\pgfmatrixcurrentcolumn+7*\pgfmatrixcurrentrow\relax->

instead.