\\
            &   &   & 0 & 0 \\
         &  &  & & 0 \\
      \end{array}
    \right)$$respectively, then $B_n$ and $B_n'$ are diffeomorphic.

Note that this lemma can be seen by the fact that $B_n$ and $B_n'$ are diffeomorphic if two associated matrices are conjugate by a permuatition matrix, see [@ch-ma-su08] or [@MP]. However, here we give a direct proof of the lemma.

Let $B$ be the $(n-2)$-stage Bott manifold with the associated $(n-2)\times (n-2)$ matrix $$\left(
      \begin{array}{ccc}
         0 & \ast & \ast  \\
         & \ddots & \ast \\
         &  & 0 \\
      \end{array}
    \right).$$ Then $B= B_{n-2} \rightarrow B_{n-3} \rightarrow \cdots \rightarrow B_1 \rightarrow B_0$, where $B_{j} = P({\mathbb{C}}\oplus \xi_{j-1}) \rightarrow B_{j-1}$.

Let $\gamma_j$ be the pull-back of the tautological line bundle of $P({\mathbb{C}}\oplus \xi_{j-1})= B_j$ via the projection $B=B_{n-2}\to B_j$, and let $c_1(\gamma_j)=x_j$ for $j \le n-2$. Let $\alpha=\sum_{i=1}^{n-2}a_ix_i$ and $\beta=\sum_{i=1}^{n-2}b_i x_i \in H^2(B)$. Define two complex line bundles over $B$ $$\gamma^\alpha=\bigotimes_{i=1}^{n-2}\gamma_i^{a_i} \rightarrow B, \text{ and } \gamma^\beta=\bigotimes_{i=1}^{n-2}\gamma_i^{b_i} \rightarrow B.$$ Let $\pi_\alpha : P({\mathbb{C}}\oplus \gamma^\alpha) \rightarrow B$ be the projection of the $\mathbb CP^1$-bundle over $B$ and denote this fiber bundle by $\eta_{\alpha}$. Similarly $\pi_\beta : P({\mathbb{C}}\oplus \gamma^\beta) \rightarrow B$ and $\eta_{\beta}$ is defined. Then $$\xymatrix{
    B_n' \cong \pi_\alpha^\ast(\eta_{\beta}) 