diff --git a/Gram-matrix-parameterization.md b/Gram-matrix-parameterization.md index 8ee20ba..df61bbc 100644 --- a/Gram-matrix-parameterization.md +++ b/Gram-matrix-parameterization.md @@ -64,12 +64,12 @@ It follows that Since the subspace $C$ is transpose-invariant, we also have \[ \mathcal{P}(X^\top) = \mathcal{P}(X)^\top. \] We can now see that -\[ +```math \begin{align*} -d\Delta & = -\mathcal{P}(dA^\top Q A + A^\top Q\,dA) \\ -& = -\big[\mathcal{P}(A^\top Q\,dA)^\top + \mathcal{P}(A^\top Q\,dA)\big]. + d\Delta & = -\mathcal{P}(dA^\top Q A + A^\top Q\,dA) \\ + & = -\big[\mathcal{P}(A^\top Q\,dA)^\top + \mathcal{P}(A^\top Q\,dA)\big]. \end{align*} -\] +``` Plugging this into our formula for $df$, and recalling that $\Delta$ is symmetric, we get ```math \begin{align*}