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-rw-r--r--execsumm/document.tex21
1 files changed, 19 insertions, 2 deletions
diff --git a/execsumm/document.tex b/execsumm/document.tex
index f3cf093..97270f1 100644
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@@ -34,14 +34,31 @@ $$-\lambda{(\lambda-{-b+\sqrt{b^2-4c}\over2})
% - 2C_1C_2\rload(3R_1+\rload).$$
Let $r_1$ and $r_2$ designate these two non-zero roots.
+The trivial zero eigenvalue corresponds to a unit x-direction vector
+by inspection. The two remaining roots, in the general case of a non-%
+degenerate system, which hasn't been explicitly ruled out, the middle
+row can be ``ignored'' because it is linearly independent in the
+following system:
+$$
+\pmatrix{-r&-C_2\rload &0\cr
+ * &* &*\cr
+ 0 &C_2R_1 &-r-C_1R_1}
+$$
+With $x = 1$, $\displaystyle y = -{r\over C_2\rload}$ and
+$\displaystyle z = y{C_2R_1\over r+C_1R_1}.$
+
\def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}}
\noindent The eigenvalues and respective eigenvectors are:
\bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$
-\bu $\lambda_2 = {r_1\over C_1C_2R_1\rload}, v_2 = ??$
+\def\num#1{\pmatrix{1\cr
+ -{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 $\lambda_3 = {r_2\over C_1C_2R_1\rload}, v_3 = ??$
+\bu $\displaystyle\lambda_3 = {r_2\over C_1C_2R_1\rload}, v_3 = \num2.$
\section{Nonhomogenous System}