\[
...\]
initiates a display math environment, which you don't need inside the equation
environment, since you're already inside math mode.

\documentclass{article}
\begin{document}
\begin{equation}
X'X = (V_r, V_{p-r}) \left( \begin{array}{cc}
\Lambda_r & 0 \\
0 & \Lambda_{p-r}
\end{array} \right) \left( \begin{array}{c}
V'_r \\
V'_{p-r}
\end{array} \right)
\end{equation}
\end{document}
Also see Where is the \matrix
command? for more ways of typesetting matrices.
Here is a minor adjustment that uses the \biggl
/\biggr
pair rather than \left
/\right
, and also reduces the outer column specification:

\documentclass{article}
\begin{document}
\begin{equation}
X'X = (V_r, V_{p-r}) \biggl( \begin{array}{@{}cc@{}}
\Lambda_r & 0 \\
0 & \Lambda_{p-r}
\end{array} \biggr) \biggl( \begin{array}{@{}c@{}}
V'_r \\
V'_{p-r}
\end{array} \biggr)
\end{equation}
\end{document}