From b68e4b382d5e09827d765038395bdad76de7959d Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Tue, 14 Apr 2020 13:21:43 -0400
Subject: removed false eigen* and fixed matrix rep

---
 execsumm/document.tex | 20 +++++++++-----------
 1 file changed, 9 insertions(+), 11 deletions(-)

diff --git a/execsumm/document.tex b/execsumm/document.tex
index c308d05..45d6546 100644
--- a/execsumm/document.tex
+++ b/execsumm/document.tex
@@ -1,5 +1,4 @@
 \def\rload{R_{\rm load}}
-\def\opt#1{\vskip0pt plus #1\vskip 0pt plus -#1}
 \input ../format
 \titlesub{Part 3: Executive Summary}{Mystery Circuit}
 
@@ -11,14 +10,15 @@ To determine the relevant properties of the linear system, matrix form
 is useful (this form was chosen to reduce fractions' usage):
 \def\x{{\bf x}}
 $$\x' =
-{1\over R_1C_1C_2\rload}\left(
+{1\over R_1C_1C_2\rload}
 \pmatrix{0&-C_2\rload      &0 \cr
          0&-C_2(R_1+\rload)&-C_1\rload\cr
          0&C_2R_1          &-C_1\rload}   \x +
+{1\over R_1}
 \pmatrix{\omega\cos(\omega t)\cr
          \omega\cos(\omega t)\cr
          0}
-\right).
+.
 $$
 The characteristic polynomial is
 $$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1\rload)
@@ -30,21 +30,19 @@ In terms of its roots (with $b=C_2(R_1+\rload)+C_1\rload$ and
 $c = C_1C_2\rload(2R_1+\rload),$
 $$-\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.
 
 \def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}}
-\opt{.15fil}
 \noindent The eigenvalues and respective eigenvectors are:
 
 \bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$
 
-\def\num{\pmatrix{C_2\rload\cr C_2\rload\cr C_2R_1}}
-\bu $\lambda_2 = -C_2\rload, v_2 = \num$
+\bu $\lambda_2 = r_1, v_2 = ??$
 
-\bu $\lambda_3 = -C_2\rload, v_2 = \pmatrix{0\cr0\cr1}$
-
-This gives the solution to the homogenous system
-$$D_1\pmatrix{1\cr0\cr0} + D_2e^{-C_2\rload t}\num
-+ D_3te^{-C_2\rload t}\pmatrix{0\cr0\cr1}.$$
+\bu $\lambda_3 = r_2, v_3 = ??$
 
 Extending to the nonhomogenous system, 
 
-- 
cgit