\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
(%i1) kill(all)$
load(eigen)$

appendix:complex eigenvalues
[email protected],26/03/2017

see:
Lecture 26: Continuation: Repeated Real Eigenvalues

(%i2) depends(x,t);
\[\tag{\%{}o2}\label{o2} [\operatorname{x}(t)]\]

simple example:

(%i4) p[1]:'diff(x[1],t)=x[1]+2*x[2];
p[2]:'diff(x[2],t)=-x[1]-x[2];
\[\tag{\%{}o3}\label{o3} \frac{d}{dt}{{x}_{1}}=2{{x}_{2}}+{{x}_{1}}\] \[\tag{\%{}o4}\label{o4} \frac{d}{dt}{{x}_{2}}=-{{x}_{2}}-{{x}_{1}}\]
(%i6) M1:zeromatrix(2,2)$
for i:1 thru 2 do
 for j:1 thru 2 do
M1[i,j]:coeff(part(p[i],2),x[j]);
\[\tag{\%{}o6}\label{o6} \mathit{done}\]

coef.Marix : 'p[i] , i=1..3'

(%i7) M1;
\[\tag{\%{}o7}\label{o7} \begin{pmatrix}1 & 2\\ -1 & -1\end{pmatrix}\]

eigenvalues ,M2       '|A-lambda*I|=0'
I=matrix(M3)

(%i11) M2:zeromatrix(2,2)$
M3:zeromatrix(2,2)$
for i:1 thru 2 do
 M3[i,i]:1;
for i:1 thru 2 do
 for j:1 thru 2 do
 M2[i,j]:M1[i,j]-lambda*M3[i,j];
\[\tag{\%{}o10}\label{o10} \mathit{done}\] \[\tag{\%{}o11}\label{o11} \mathit{done}\]
(%i12) M2;
\[\tag{\%{}o12}\label{o12} \begin{pmatrix}1-\mathit{lambda} & 2\\ -1 & -\mathit{lambda}-1\end{pmatrix}\]

find :eigenvalues 'lambda'

(%i13) p1:ratsimp(determinant(M2))=0;
\[\tag{p1}\label{p1}{{\mathit{lambda}}^{2}}+1=0\]
(%i18) p2:allroots(p1);
p21:p2[1];
p22:p2[2];
p210:part(p21,2);
p220:part(p22,2);
\[\tag{p2}\label{p2}[\mathit{lambda}=1.0\%{}i,\mathit{lambda}=-1.0\%{}i]\] \[\tag{p21}\label{p21}\mathit{lambda}=1.0\%{}i\] \[\tag{p22}\label{p22}\mathit{lambda}=-1.0\%{}i\] \[\tag{p210}\label{p210}1.0\%{}i\] \[\tag{p220}\label{p220}-1.0\%{}i\]
(%i19) p3:[vals,vecs]:eigenvectors(M1);
\[\tag{p3}\label{p3}[[[-\%{}i,\%{}i],[1,1]],[[[1,-\frac{\%{}i+1}{2}]],[[1,\frac{\%{}i-1}{2}]]]]\]
(%i21) p4:vals[1];
p5:vecs;
\[\tag{p4}\label{p4}[-\%{}i,\%{}i]\] \[\tag{p5}\label{p5}[[[1,-\frac{\%{}i+1}{2}]],[[1,\frac{\%{}i-1}{2}]]]\]
(%i23) p6:p5[1][1];
p7:p5[2][1];
\[\tag{p6}\label{p6}[1,-\frac{\%{}i+1}{2}]\] \[\tag{p7}\label{p7}[1,\frac{\%{}i-1}{2}]\]

solution 'system diff.equations':p[i],i=1,2,3
math: e^(i*t) = cos(t)+i*sin(t)

(%i25) p8:x[1]=c1*p7[1]*%e^(p210*t)+c2*p6[1]*%e^(p220*t);
p9:x[2]=c1*p7[2]*%e^(p210*t)+c2*p6[2]*%e^(p220*t);;
\[\tag{p8}\label{p8}{{x}_{1}}=\mathit{c1}\,{{\%{}e}^{1.0\%{}it}}+\mathit{c2}\,{{\%{}e}^{-1.0\%{}it}}\] \[\tag{p9}\label{p9}{{x}_{2}}=\frac{\left( \%{}i-1\right) \,\mathit{c1}\,{{\%{}e}^{1.0\%{}it}}}{2}-\frac{\left( \%{}i+1\right) \,\mathit{c2}\,{{\%{}e}^{-1.0\%{}it}}}{2}\]

'x '(t)
   1       1

realpart 'x '
          1

(%i26) p10:realpart(part(p8,2));
\[\tag{p10}\label{p10}\mathit{c2}\,\cos{\left( 1.0t\right) }+\mathit{c1}\,\cos{\left( 1.0t\right) }\]

imaginairpart 'x '
               1

(%i27) p11:imagpart(part(p8,2));
\[\tag{p11}\label{p11}\mathit{c1}\,\sin{\left( 1.0t\right) }-\mathit{c2}\,\sin{\left( 1.0t\right) }\]

'x '(t)
  2

realpart 'x '
          2

(%i28) p12:realpart(part(p9,2));
\[\tag{p12}\label{p12}\frac{\mathit{c1}\,\left( -\sin{\left( 1.0t\right) }-\cos{\left( 1.0t\right) }\right) }{2}-\frac{\mathit{c2}\,\left( \sin{\left( 1.0t\right) }+\cos{\left( 1.0t\right) }\right) }{2}\]

imaginairpart 'x '
               2

(%i29) p13:imagpart(part(p9,2));
\[\tag{p13}\label{p13}\frac{\mathit{c1}\,\left( \cos{\left( 1.0t\right) }-\sin{\left( 1.0t\right) }\right) }{2}-\frac{\mathit{c2}\,\left( \cos{\left( 1.0t\right) }-\sin{\left( 1.0t\right) }\right) }{2}\]

statement : imaginair numbers = realpart+%i*imagpart

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