space emerging from the classical solutions in more detail, we define an $n \times n$ matrix $$Q\left(t\right)=\sum_{i=1}^{3} \bar{X}_{i}(t)^2  \ ,$$ whose eigenvalues $q_{r}(t)$ represent the radial distribution of the points which describe the 3-dimensional space. Note that the eigenvalues $q_{r}(t)$ are related to the extent of space $R^{2}(t)$ as $$R^{2} (t) 
=\frac{1}{n}\sum_{r=1}^{n} q_{r} (t) \ .$$ In Fig. \[fig:Q\], we plot the eigenvalues of $Q(t)$ for the same typical solution discussed above. We find that they are densely distributed at each time and that all the eigenvalues are growing with time in the $t>0$ region. This is in sharp contrast to the results for the configurations obtained by the previous Monte Carlo studies of the Lorentzian type IIB matrix model [@Aoki:2019tby], where only two of the eigenvalues grow with time, while the others remain small and constant. As we mentioned in the Introduction, the latter behavior is interpreted as a consequence of the approximation used in the Monte Carlo studies to avoid the sign problem. Our results for the classical solutions suggest the emergence of *smooth* (3+1)-dimensional expanding space-time in the large-$N$ limit.

The emergence of intersecting D-branes {#sec: results_chiralzeromode}
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In this section we discuss how chiral zero modes 