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diff --git a/execsumm/document.tex b/execsumm/document.tex index f3cf093..97270f1 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -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} |