From 76781234f2c320ca71f308ada5a2f5b4d9154e59 Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Tue, 14 Apr 2020 20:32:26 -0400
Subject: update with a correction and more clarity

---
 execsumm/document.tex | 16 +++++++++-------
 1 file changed, 9 insertions(+), 7 deletions(-)

diff --git a/execsumm/document.tex b/execsumm/document.tex
index 95146a4..930705b 100644
--- a/execsumm/document.tex
+++ b/execsumm/document.tex
@@ -12,7 +12,7 @@ is useful (this form was chosen to reduce fractions' usage):
 $$\x' =
 {1\over R_1C_1C_2\rload}
 \pmatrix{0&-C_2\rload      &0 \cr
-         0&-C_2(R_1+\rload)&-C_1R_1\cr
+         0&-C_2(R_1+\rload)&C_1R_1\cr
          0&C_2R_1          &-C_1R_1}   \x +
 {1\over R_1}
 \pmatrix{\omega\cos(\omega t)\cr
@@ -21,18 +21,20 @@ $$\x' =
 .$$
 The characteristic polynomial is
 $$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1R_1)
-+ C_2C_1R_1^2).$$
+- C_2C_1R_1^2).$$
 Expanded,
 $$-\lambda( \lambda^2 + \lambda(C_2(R_1+\rload)+C_1R_1)
-+ C_1C_2R_1(2R_1+\rload)).$$
++ C_1C_2R_1\rload).$$
 In terms of its roots (with $b=C_2(R_1+\rload)+C_1R_1$ and
-$c = C_1C_2R_1(2R_1+\rload),$
+$c = C_1C_2R_1\rload,$
 $$-\lambda{(\lambda-{-b+\sqrt{b^2-4c}\over2})
 (\lambda-{-b-\sqrt{b^2-4c}\over 2})}$$
 %For reference,
 %$$b^2-4c = C_2^2(R_1+\rload)^2 + C_1^2\rload^2
 %           - 2C_1C_2\rload(3R_1+\rload).$$
 Let $r_1$ and $r_2$ designate these two non-zero roots.
+In the original matrix $A$, this being the transformed matrix $A/c$,
+$Av = cr_1$ because $A/c * v = r_1.$
 
 The trivial zero eigenvalue corresponds to a unit x-direction vector
 by inspection. The two remaining roots, in the general case of a non-%
@@ -56,9 +58,9 @@ $\displaystyle z = y{C_2R_1\over r+C_1R_1}.$
                     -{r_#1\over C_2\rload}\cr
                     -{R_1r_#1\over \rload(r_#1+C_1R_1)}}}
 
-\bu $\displaystyle\lambda_2 = {r_1\over C_1C_2R_1\rload}, v_2 = \num1.$
+\bu $\displaystyle\lambda_2 = {r_1C_1C_2R_1\rload}, v_2 = \num1.$
 
-\bu $\displaystyle\lambda_3 = {r_2\over C_1C_2R_1\rload}, v_3 = \num2.$
+\bu $\displaystyle\lambda_3 = {r_2C_1C_2R_1\rload}, v_3 = \num2.$
 
 This corresponds to a solution of the form $f = Ce^{\lambda t}v,\cdots$
 where $C\in {\bf C},$ $f\in {\bf R}\to{\bf R}$. If the non-trivial
@@ -72,6 +74,6 @@ $g = C_1\cos{\re(\lambda)t}\re v + C_2\sin{\re(\lambda)t}\im v.$ %% DOUBLE CHECK
 Extending to the nonhomogeneous system will take slightly different
 paths depending on if the system has complex roots or has real roots.
 But in either case, $\cos x*{\rm polynomial}+\sin x*{\rm polynomial}$
-should be a particular solution
+should be a particular solution.
 
 \bye
-- 
cgit