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update notations

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Trance-0
2025-11-04 12:43:23 -06:00
parent d24c0bdd9e
commit 614479e4d0
27 changed files with 333 additions and 100 deletions
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content/CSE559A/CSE559A_L22.md
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@@ -178,7 +178,7 @@ $$
\begin{pmatrix}a\\b\\c\end{pmatrix} \times \begin{pmatrix}a'\\b'\\c'\end{pmatrix} = \begin{pmatrix}bc'-b'c\\ca'-c'a\\ab'-a'b\end{pmatrix}
$$
Let $h_1^T, h_2^T, h_3^T$ be the rows of $H$. Then
Let $h_1^\top, h_2^\top, h_3^\top$ be the rows of $H$. Then
$$
x_i' × Hx_i=\begin{pmatrix}
@@ -186,15 +186,15 @@ x_i' × Hx_i=\begin{pmatrix}
y_i' \\
1
\end{pmatrix} \times \begin{pmatrix}
h_1^T x_i \\
h_2^T x_i \\
h_3^T x_i
h_1^\top x_i \\
h_2^\top x_i \\
h_3^\top x_i
\end{pmatrix}
=
\begin{pmatrix}
y_i' h_3^T x_i−h_2^T x_i \\
h_1^T x_i−x_i' h_3^T x_i \\
x_i' h_2^T x_i−y_i' h_1^T x_i
y_i' h_3^\top x_i−h_2^\top x_i \\
h_1^\top x_i−x_i' h_3^\top x_i \\
x_i' h_2^\top x_i−y_i' h_1^\top x_i
\end{pmatrix}
$$
@@ -206,15 +206,15 @@ x_i' × Hx_i=\begin{pmatrix}
y_i' \\
1
\end{pmatrix} \times \begin{pmatrix}
h_1^T x_i \\
h_2^T x_i \\
h_3^T x_i
h_1^\top x_i \\
h_2^\top x_i \\
h_3^\top x_i
\end{pmatrix}
=
\begin{pmatrix}
y_i' h_3^T x_i−h_2^T x_i \\
h_1^T x_i−x_i' h_3^T x_i \\
x_i' h_2^T x_i−y_i' h_1^T x_i
y_i' h_3^\top x_i−h_2^\top x_i \\
h_1^\top x_i−x_i' h_3^\top x_i \\
x_i' h_2^\top x_i−y_i' h_1^\top x_i
\end{pmatrix}
$$
@@ -222,9 +222,9 @@ Rearranging the terms:
$$
\begin{bmatrix}
0^T &-x_i^T &y_i' x_i^T \\
x_i^T &0^T &-x_i' x_i^T \\
y_i' x_i^T &x_i' x_i^T &0^T
0^\top &-x_i^\top &y_i' x_i^\top \\
x_i^\top &0^\top &-x_i' x_i^\top \\
y_i' x_i^\top &x_i' x_i^\top &0^\top
\end{bmatrix}
\begin{bmatrix}
h_1 \\