From ee6276f9264bc8585dd3600eff94e7407cca4f92 Mon Sep 17 00:00:00 2001 From: Vectornaut Date: Mon, 18 Nov 2024 22:57:04 +0000 Subject: [PATCH] Correct sign error --- Gram-matrix-parameterization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Gram-matrix-parameterization.md b/Gram-matrix-parameterization.md index f38e8b5..e8d7cfc 100644 --- a/Gram-matrix-parameterization.md +++ b/Gram-matrix-parameterization.md @@ -101,7 +101,7 @@ we can express the derivative of $\operatorname{grad}(f)$ as \[ \begin{align*} d\operatorname{grad}(f) & = -4 Q\,dA\,\mathcal{P}(\Delta) - 4 Q A\,\mathcal{P}(d\Delta) \\ -& = -4 Q\big[dA\,\mathcal{P}(\Delta) + A\,\mathcal{P}(-d\Delta)\big]. +& = 4 Q\big[{-dA}\,\mathcal{P}(\Delta) + A\,\mathcal{P}(-d\Delta)\big]. \end{align*} \] In the Rust and Julia implementations of the realization routine, we express $d\operatorname{grad}(f)$ as a matrix in the standard basis for $\operatorname{End}(\mathbb{R}^n)$. We apply the cotangent vector $d\operatorname{grad}(f)$ to each standard basis matrix $E_{ij}$ by setting the value of the matrix-valued 1-form $dA$ to $E_{ij}$.