Spacing in matrix entities

I will try to get something like in the picture below:

I have tried a code but the spacing are quite large compared to the spacing in the picture

$$\begin{matrix} \xi_1 & = & \xi_{10} & + & \xi_{11}\,X_1 & + & \cdots & + & \xi_{1n}\,X_n\\ \xi_2 & = & \xi_{20} & + & \xi_{21}\,X_1 & + & \cdots & + & \xi_{2n}\,X_n\\ & \vdots & & & & & \vdots & & \\ \xi_n & = & \xi_{n0} & + & \xi_{11}\,X_1 & + & \cdots & + & \xi_{nn}\,X_n\\ \end{matrix}$$


The code here gives the following

Using a matrix environment doesn't seem right. Here's an array-based solution.

I think it's neither necessary nor (typographically) helpful to insert \, (thinspace) between the \xi and X terms. For sure, your upper screenshot doesn't appear to indicate any such whitespace ought to be inserted.

\documentclass{article}
\usepackage{array}
\newcolumntype{C}{>{{}}c<{{}}}

\begin{document}
$$\setlength\arraycolsep{0pt} \begin{array}{rCrCrCrCr} \xi_1 & = & \xi_{10} & + & \xi_{11}X_1 & + & \cdots & + & \xi_{1n}X_n\\ \xi_2 & = & \xi_{20} & + & \xi_{21}X_1 & + & \cdots & + & \xi_{2n}X_n\\ & \vdots & & & & & & \vdots & \\ \xi_n & = & \xi_{n0} & + & \xi_{11}X_1 & + & \cdots & + & \xi_{nn}X_n\\ \end{array}$$
\end{document}

$$\setlength\arraycolsep{1.6pt} \begin{matrix} \xi_1 & = & \xi_{10} & + & \xi_{11}\,X_1 & + & \cdots & + & \xi_{1n}\,X_n\\ \xi_2 & = & \xi_{20} & + & \xi_{21}\,X_1 & + & \cdots & + & \xi_{2n}\,X_n\\ & \vdots & & & & & \vdots & & \\ \xi_n & = & \xi_{n0} & + & \xi_{11}\,X_1 & + & \cdots & + & \xi_{nn}\,X_n\\ \end{matrix}$$


I think using alignat will be simpler:

\documentclass{article}
\usepackage{mathtools}
\begin{document}

\begin{alignat*}{4}
\xi_1 & = \xi_{10} & & + \xi_{11}X_1 & & + \cdots & & + \xi_{1n}X_n\\
\xi_2 & = \xi_{20} & & + \xi_{21}X_1 & & + \cdots & & + \xi_{2n}X_n\\
& \vdotswithin{=} & & & & \vdotswithin{+}
\\
\xi_n & = \xi_{n0} & & + \xi_{11}X_1 & & + \cdots & & + \xi_{nn}X_n
\end{alignat*}

\end{document}