p \cdot \hat{X}_\mu$, where $$p =
\left(
\begin{matrix}
 &       &1& & &   \\
 & {\reflectbox{$\ddots$}}& & &\mbox{\Huge $0$}&  \\
1&       & & & &   \\
 & &       & &       &1   \\
 &\mbox{\Huge $0$}&       & &{\reflectbox{$\ddots$}}&   \\
 & &       &1&       &    \\
\end{matrix}
\right)$$ This is the tangent space analogue of intersecting a Schubert variety with an opposite Schubert variety.

\[ex:oppositeintersection\] If $\lambda = \mathbf{0} \leq 0 \leq 1 \leq 3 \leq 3 \leq \mathbf{5}$, $\mu = \mathbf{0} \leq 3 \leq 3 \leq 3 \leq 5 \leq \mathbf{5}$, then $$\begin{aligned}
X_\lambda \cap X_\mu &=
\left (
\begin{matrix}
0 & * & * & * \\
0 & 0 & * & * \\
0 & 0 & * & * \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 
\end{matrix}
\right)
\cap
\left (
\begin{matrix}
* & 0 & 0 & 0 \\
* & 0 & 0 & 0 \\
* & * & * & * \\
* & * & * & * \\
* & * & * & * 
\end{matrix}
\right) \\
&= 
\left (
\begin{matrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 