From c98626855901bab52bb7758177651e657fa70d8e Mon Sep 17 00:00:00 2001
From: YigitElma <yigitelmacioglu@gmail.com>
Date: Sun, 20 Oct 2024 23:28:06 -0400
Subject: [PATCH] fix more errors

---
 docs/dev_guide/transform_2.ipynb | 15 ++++++++++++---
 1 file changed, 12 insertions(+), 3 deletions(-)

diff --git a/docs/dev_guide/transform_2.ipynb b/docs/dev_guide/transform_2.ipynb
index 82f4bf2d8c..ab985265b5 100644
--- a/docs/dev_guide/transform_2.ipynb
+++ b/docs/dev_guide/transform_2.ipynb
@@ -52,6 +52,7 @@
     "\n",
     "Define the transform matrix as $A_{(d\\rho,d\\theta,d\\zeta)}$ for the derivative of order ${(d\\rho,d\\theta,d\\zeta)}$ (where each are integers).\n",
     "This matrix transforms a spectral basis evaluated on a certain grid with a given set of coefficients $\\mathbf{c}$ to real space values $x$.\n",
+    "\n",
     "$$ A\\mathbf{c} = \\mathbf{x}$$\n",
     "\n",
     "- $\\mathbf{c}$ is a vector of length `Transform.basis.num_modes` (the number of modes in the basis)\n",
@@ -93,18 +94,26 @@
     "The pseudo-inverse transform, $A^{\\dagger}$, applied to $\\mathbf{x}$ represents the least-squares solution for the unknown given by $\\mathbf{c}$ to the system $A \\mathbf{c} = \\mathbf{x}$.\n",
     "\n",
     "It is required from the least-squares solution, $A^{\\dagger} \\mathbf{x}$, that\n",
+    "\n",
     "$$A^{\\dagger} \\mathbf{x} = \\min_{∀ \\mathbf{c}} \\lvert A \\mathbf{c} - \\mathbf{x} \\rvert \\; \\text{so that} \\; \\lvert A A^{\\dagger} \\mathbf{x} - \\mathbf{x}\\rvert \\; \\text{is minimized}$$\n",
+    "\n",
     "For this to be true, $A A^{\\dagger}$ must be the orthogonal projection onto the image of the transformation $A$.\n",
     "It follows that\n",
+    "\n",
     "$$A A^{\\dagger} \\mathbf{x} - \\mathbf{x} ∈ (\\text{image}(A))^{\\perp} = \\text{kernel}(A^T)$$\n",
+    "\n",
+    "$$\n",
     "\\begin{align*}\n",
     "    A^T (A A^{\\dagger} \\mathbf{x} - \\mathbf{x}) &= 0 \\\\\n",
     "    A^T A A^{\\dagger} \\mathbf{x} &= A^T \\mathbf{x} \\\\\n",
     "    A^{\\dagger} &= (A^T A)^{-1} A^{T} \\quad \\text{if} \\; A \\; \\text{is invertible}\n",
     "\\end{align*}\n",
+    "$$\n",
     "\n",
     "Equivalently, if $A = U S V^{T}$ is the singular value decomposition of the transform matrix $A$, then\n",
+    "\n",
     "$$ A^{\\dagger} = V S^{+} U^{T}$$\n",
+    "\n",
     "where the diagonal of $S^{+}$ has entries which are are the recipricols of the entries on the diagonal of $S$, except that any entries in the diagonal with $0$ for the singular value are kept as $0$.\n",
     "(If there are no singular values corresponding to $0$, then $S^{+}=S^{-1} \\implies A^{\\dagger}=A^{-1}$, and hence $A^{-1}$ exists because there are no eigenvectors with eigenvalue $0^{2}$)."
    ]
@@ -170,13 +179,13 @@
     "Functions of the toroidal coordinate $\\zeta$ use Fourier series for their basis.\n",
     "So a Fourier transform can be used to transform real space values to spectral space for the pseudoinverse matrix.\n",
     "\n",
-    "Todo: Figure out how fft algorithm is used."
+    "`Todo: Figure out how fft algorithm is used.`"
    ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
+   "display_name": "desc-env",
    "language": "python",
    "name": "python3"
   },
@@ -190,7 +199,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.16"
+   "version": "3.12.0"
   }
  },
  "nbformat": 4,