asciidoc -> qmd

docs/content/notes/danilevsky.qmd

@@ -358,7 +358,7 @@ a_{kj}   &=
 \end{cases}
 \end{aligned}
 $$
-That last step puts row $k$ of $A$ into companion matrix form--the later rows of $A$ are already there.
+That last step puts row $k$ of $A$ into companion matrix form --- the later rows of $A$ are already there.
 
 .. If $k > 1$, then $k \leftarrow k-1$, and we go back to step 1. Otherwise, we are finished.
 
@@ -421,7 +421,7 @@ And of course as usual, the additive inverse of 0 is 0.
 This means that in $\mathbb{F_2}$ negation is a no-op and any term like $-b$ can just be replaced with $b$.
 
 We always have $1 * 1 = 1$ so the multiplicative inverse of $1$ is just $1$.
-Also just like $\mathbb{R}$, the element $0$ has no multiplicative inverse in $\mathbb{F_2}$ either--you still cannot divide by zero.
+Also just like $\mathbb{R}$, the element $0$ has no multiplicative inverse in $\mathbb{F_2}$ either --- you still cannot divide by zero.
 This means that if $a, b \in \mathbb{F_2}$ then a term like $a/b$ makes no sense if $b=0$ but otherwise $a/b = a$.
 
 Let's reconsider that very first step we took above to move our matrix $A$ closer to Frobenius form.
@@ -452,20 +452,19 @@ As before, it proceeds in a sequence of steps that move $A$ closer to companion/
 
 ### The Algorithm
 
-. Initialize a counter $k$ to $n$ and $A$ to the input matrix.
+1. Initialize a counter $k$ to $n$ and $A$ to the input matrix.
 
-. If $a_{kk-1} = 0$ look for an earlier element in the same row that is 1.
-That is, look for $j < k-1$ where $a_{kj} = 1$. +
-If we are successful then swap both the _rows and columns_ $j$ and $k-1$ of $A$. +
-Row and column swaps like that are permutation transformations and those preserve the eigen-structure.
+2. If $a_{kk-1} = 0$ look for an earlier element in the same row that is 1. That is, look for an index $j < k-1$ such that $a_{kj} = 1$. If found then swap *both* the rows and the columns $j$ and $k-1$ of $A$. These swaps are permutation transformations that preserve the eigen-structure.
 
-. If by now $a_{kk-1} = 1$:
+3. If after step 2 we have $a_{kk-1} = 1$:
 
-.. Capture the elements from row $k$ of the matrix $A$ by setting $m_ = \text{row}_k(A)$. +
-Note that by assumption $m_{k-1} = a_{kk-1} = 1$.
+    + Capture the elements from row $k$ of the matrix $A$ by setting
+$$
+    m = \text{row}_k(A).
+$$
+      Note that by assumption $m_{k-1} = a_{kk-1} = 1$.
 
-.. Compute the elements of $B$ for rows $i = 1, ..., k-1$ as follows:
-+
+    + Compute the elements of $B$ for rows $i = 1, ..., k-1$ as follows
 $$
 b_{ij} =
 \begin{cases}
@@ -474,8 +473,7 @@ b_{ij} =
 \end{cases}
 $$
 
-.. Update $A$ for all columns $j = 1, \ldots, n$ as follows:
-+
+    + Update $A$ for all columns $j = 1, \ldots, n$ as follows
 $$
 \begin{aligned}
 a_{ij}   &=  b_{ij} \text{ for } i = 1, \ldots, k-2, \\
@@ -487,36 +485,29 @@ a_{kj}   &=
 \end{cases}
 \end{aligned}
 $$
-+
-That last step puts row $k$ of $A$ into Frobenius form--the later rows of $A$ are already there.
+That last step puts row $k$ of $A$ into Frobenius form --- later rows are already there.
 
-.. If $k > 1$ then $k \leftarrow k-1$ and we go back to step 1, otherwise we are done.
+    + If $k > 1$ then $k \gets k-1$ and we go back to step 1, otherwise, we are done.
 
-. If $a_{kk-1} = 0$ even after trying the search in step 2 then we cannot perform step 3. +
-Just like before, in this case the current $A$ matrix must have the following form:
-+
+4. If after step 2 we have $a_{kk-1} = 0$ then we cannot perform step 3. \
+In this case, the current $A$ matrix must have the following form:
 $$
 A =
 \begin{bmatrix}
 A_1 & A_2 \\
 0   & A_3
 \end{bmatrix}
 $$
-hence the characteristic polynomial $c_A(x)$ we are after is the product of two other characteristic polynomials:
-+
+hence the characteristic polynomial $c_A(x)$ is the product of two others: \
 $$
-c_A(x) = c_{A_1}(x) c_{A_3}(x)
+c_A(x) = c_{A_1}(x) c_{A_3}(x).
 $$
-+
-and as $A_3$ is already in Frobenius form we can easily read off the coefficients for $c_{A_3}(x)$:
-+
+As $A_3$ is already in Frobenius form we can easily read off the coefficients for $c_{A_3}(x)$: \
 $$
 c_{A_3}(x) = 1+ a_{kk} x + a_{kk+1} x^2 + \ldots + a_{kn} x^{n-k+1}.
 $$
-+
-The algorithm will store those coefficients and recurse using $A_1$ as the smaller $(k-1) \times (k-1)$  input matrix.
-+
-It can then convolve the coefficients of $c_{A_1}(x)$ and $c_{A_3}(x)$ to return the coefficients for $c_{A}(x)$.
+Store those coefficients and recurse using $A_1$ as the $(k-1) \times (k-1)$ input matrix.
+Convolve the coefficients of $c_{A_1}(x)$ and $c_{A_3}(x)$ to get the coefficients for $c_{A}(x)$.
 
 ::: {.callout-note}
 # Recursion is to be expected here