fixed error and added characteristic polynomials

1 files changed, 13 insertions, 7 deletions

diff --git a/execsumm/document.tex b/execsumm/document.tex
index b5dfdc7..32d4a0a 100644
--- a/execsumm/document.tex
+++ b/execsumm/document.tex
@@ -8,20 +8,26 @@
 \section{Matrix Representation}
 
 To determine the relevant properties of the linear system, matrix form
-is useful:
+is useful (this form was chosen to reduce fractions' usage):
 \def\x{{\bf x}}
 $$\x' =
 {1\over R_1C_1C_2\rload}\left(
-\pmatrix{0&-C_2\rload&0 \cr
-         *&*         &* \cr
-         0&C_2R_1    &-C_2\rload}   \x +
+\pmatrix{0&-C_2\rload      &0 \cr
+         0&-C_2(R_1+\rload)&-C_1\rload\cr
+         0&C_2R_1          &-C_1\rload}   \x +
 \pmatrix{\omega\cos(\omega t)\cr
          \omega\cos(\omega t)\cr
          0}
 \right).
 $$
-
-{\it *Linearly dependent, meaning a trivial eigenvalue of 0}
+The characteristic polynomial is
+$$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1\rload)
++ C_2^2R_1\rload).$$
+Expanded,
+$$-\lambda( \lambda^2 + \lambda(C_2(R_1+\rload)+C_1\rload)
++ C_1C_2\rload(R_1+\rload) + C_2^2R_1\rload).$$
+In terms of its roots,
+$$???$$
 
 \def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}}
 \opt{.15fil}
@@ -29,7 +35,7 @@ $$
 
 \bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$
 
-\def\num{\pmatrix{0\cr C_2\rload\cr C_2R_1}}
+\def\num{\pmatrix{C_2\rload\cr C_2\rload\cr C_2R_1}}
 \bu $\lambda_2 = -C_2\rload, v_2 = \num$
 
 \bu $\lambda_3 = -C_2\rload, v_2 = \pmatrix{0\cr0\cr1}$