Update Gram matrix parameterization

Gram-matrix-parameterization.md

@@ -39,7 +39,12 @@ on $\operatorname{Hom}(\mathbb{R}^n, V)$. Finding a global minimum of the *loss
 #### The first derivative of the loss function
 
 Write the loss function as
-\[ \begin{align*} f & = \|\Delta\|^2 \\ & = \operatorname{tr}(\Delta^\top \Delta), \end{align*} \]
+```math
+\begin{align*} 
+   f & = \|\Delta\|^2 \\ 
+     & = \operatorname{tr}(\Delta^\top \Delta), 
+\end{align*}
+```
 where $\Delta = G - \mathcal{P}(A^\top Q A)$. Differentiate both sides and simplify the result using the transpose-invariance of the trace:
 \[ \begin{align*} df & = \operatorname{tr}(d\Delta^\top \Delta) + \operatorname{tr}(\Delta^\top d\Delta) \\ & = 2\operatorname{tr}(\Delta^\top d\Delta). \end{align*} \]
 To compute $d\Delta$, it will be helpful to write the projection operator $\mathcal{P}$ more explicitly. Let $\mathcal{C}$ be the set of indices where the Gram matrix is unconstrained. We can express $C$ as the span of the standard basis matrices $\{E_{ij}\}_{(i, j) \in \mathcal{C}}$. Observing that $E_{ij} X^\top E_{ij} = X_{ij} E_{ij}$ for any matrix $X$, we can do orthogonal projection onto $C$ using standard basis matrices: