How do the following matrix product

Question

I am interesting about drafting the following in LaTeX. I am not so much expert in this, i am going to attach the code to generate the vector and matrix, and after that I am going to attach my desired output.

Code:

\begin{align}
  \begin{pmatrix}a_1\\a_2\\\vdots\\a_k\end{pmatrix}
  \begin{pmatrix}\varepsilon_1&0&\ldots&0\\ 0&\varepsilon_2&\ldots&0\\0&0&\ldots&0\\0&0&\ldots&\varepsilon_N\\ 
   \end{pmatrix}
\end{align}

Output:

So what I want is a circle on the first and last element of vector (a1,..,ak) and an arrow that goes from a1 to the rectangle of first row of matrix, the same for the last row. Sorry if the circle is made in that bad way, I have done it manually, of course is preferred a circle draft in a better way.

Answer 1

With {NiceArray} of nicematrix and Tikz.

\documentclass{article}
\usepackage{nicematrix,tikz}
\usetikzlibrary{fit}
\begin{document}
$\begin{NiceArray}{(c)(cccc)}[create-medium-nodes,nullify-dots]
a_1    & \epsilon_1 & 0          & \cdots & 0 \
a_2    & 0          & \epsilon_2 & \cdots & 0 \
\Vdots & 0          & 0          & \cdots & 0 \
a_k    & 0          & 0          & \cdots & \epsilon_n \
\CodeAfter
  \begin{tikzpicture} [cyan,inner sep=1pt]
    \node [circle,draw,fit=(1-1)] (A) {} ;
    \node [circle,draw,fit=(4-1)] (B) {} ;
    \node [draw,rounded corners, fit = (1-2-medium) (1-5-medium)] (C) {} ;
    \node [draw,rounded corners, fit = (4-2-medium) (4-5-medium)] (D) {} ;
    \draw [->] (A.north) to [bend left] (C.north west) ; 
    \draw [->] (B.south) to [bend right] (D.south west) ; 
  \end{tikzpicture}
\end{NiceArray}$
\end{document}

You need several compilations (because nicematrix uses PGF/Tikz nodes under the hood).

Answer 2

With pure TikZ:

\documentclass[border=3.141592]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,
                fit,
                matrix,
                positioning}
\begin{document}
    \begin{tikzpicture}[
node distance = 7mm,
FIT/.style = {inner sep=1pt, draw=red, fit=#1},
arr/.style = {draw=blue, -Straight Barb,semithick},
M/.style = {matrix of math nodes,
            nodes= {minimum size=1em, inner sep=1pt, anchor=base},
            column sep=0pt,
            row sep=3pt,
            left delimiter=(,
            right delimiter=),
            inner sep=2pt}

                        ]
\matrix (m) [M] 
{|[circle,draw=red]| a_1\
    a_2\ \vdots \ 
 |[circle,draw=red]|a_k\};
%
\matrix (n) [M,right=of m]
{\varepsilon_1  & 0             & \ldots    & 0             \
    0           &\varepsilon_2  & \ldots    & 0             \
    0           & 0             & \ldots    & 0             \
    0           & 0             & \ldots    & \varepsilon_N \
};
\node (n1) [FIT=(n-1-1) (n-1-4)] {};
\node (n2) [FIT=(n-4-1) (n-4-4)] {};
%
\draw[arr] (m-1-1) to[bend  left=45] (n1);
\draw[arr] (m-4-1) to[bend right=45] (n2);
    \end{tikzpicture}
\end{document}

Answer 3

Here is a possibility with a simple pstricks code:

\documentclass[svgnames]{article}
\usepackage{amsmath}
\usepackage{pst-node}
\begin{document}
\[ \psset{linecolor=LightSteelBlue, linearc=0.1, boxsize=0.25, nodesep=1.5pt, arrowinset=0.12} \begin{pmatrix}\circlenode{a1}{a_1}\\a_2\\\vdots\\\circlenode{ak}{a_k}\end{pmatrix}
\begin{pmatrix}\,\Rnode{e1}{\varepsilon_1}&0&\ldots&\Rnode{f1}{0}\\ \, 0 & \varepsilon_2&\ldots&0\\ \,0&0&\ldots&0\\ \,\Rnode{fN}{0}&0&\ldots&\Rnode{eN}{\varepsilon_N}\\
\end{pmatrix}
\ncbox{e1}{f1}\ncbox{eN}{fN}
\nccurve[angle=90, nodesepA=0, nodesepB=5pt]{->}{a1}{e1}
\nccurve[angle=-90, nodesepA=0, nodesepB=3pt]{->}{ak}{fN}
\]%

\end{document}

Answer 4

And now with a simple TikZ solution, using tikzmark library, as suggested in my comment (note: needs to be compiled a couple of times to compute then display correctly).

\documentclass[12pt]{article}
\usepackage{tikz,amsmath}
\usetikzlibrary{tikzmark,fit}
\begin{document}
\begin{align}
    \begin{pmatrix}\tikzmarknode{A1}{a_1}\\a_2\\\vdots\\\tikzmarknode{AK}{a_k}\end{pmatrix}
    \begin{pmatrix}
        \tikzmarknode{E1}{\varepsilon_1}&0&\ldots&\tikzmarknode{0A}{0}\\
        0&\varepsilon_2&\ldots&0\\
        0&0&\ldots&0\\
        \tikzmarknode{0B}{0}&0&\ldots&\tikzmarknode{EN}{\varepsilon_N}\\ 
    \end{pmatrix}
\end{align}

\tikzset{
    circ/.style={inner sep=6pt, outer sep=0pt, line width=1pt, circle, draw=cyan},
    rect/.style={inner sep=2pt, outer sep=0pt, line width=1pt, rounded corners=5pt, draw=cyan}
    }

\begin{tikzpicture}[remember picture,overlay,cyan]
    \node[circ] (a1) at (A1) {};
    \node[circ] (ak) at (AK) {};

    \node[rect,fit=(E1)(0A)] (line1) {};
    \node[rect,fit=(EN)(0B)] (line4) {};

    \draw[-latex,line width=1pt] (a1.north) to[out=60,in=120] (line1.160);
    \draw[-latex,line width=1pt] (ak.south) to[out=-60,in=-120] (line4.-160);

\end{tikzpicture}


\end{document}