\magnification=\magstep1

$$
{\rm matrix\ } {\bf A} \quad = \quad \pmatrix{ 2 & 2 \cr 5 & -1 \cr } 
$$

$$
{\rm vector\ } {\bf x} \quad = \quad \pmatrix{ 1 \cr 2 \cr } 
$$

$$
{\bf A x } \quad = \quad 
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ 1 \cr 2 \cr } 
     \quad = \quad 
        \pmatrix{ 2 + 4 \cr 5  -2 \cr }
     \quad = \quad 
        \pmatrix{ 6 \cr 3 \cr }
$$

$$
{\bf x } = \pmatrix{ 1 \cr 2 \cr } \quad {\rm vs\ } \quad 
    {\bf A x } = \pmatrix{ 6 \cr 3 \cr }
$$


$$
{\rm vector\ } {\bf x} \quad = \quad \pmatrix{ 1 \cr 1 \cr } 
$$

$$
{\bf A x } \quad = \quad 
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ 1 \cr 1 \cr } 
     \quad = \quad 
        \pmatrix{ 2 + 2 \cr 5  -1 \cr }
     \quad = \quad 
        \pmatrix{ 4 \cr 4 \cr }
$$

$$
{\bf x } = \pmatrix{ 1 \cr 1 \cr } \quad {\rm vs\ } \quad 
    {\bf A x } = \pmatrix{ 4 \cr 4 \cr }
$$

$$
{\bf A x } \quad = \quad \pmatrix{4 \cr 4 \cr } \quad = \quad
       4 {\bf x }
$$

$$
{\rm vector\ } {\bf x} \quad = \quad \pmatrix{ 2 \cr 2 \cr } 
$$

$$
{\bf A x } \quad = \quad 
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ 2 \cr 2 \cr } 
     \quad = \quad 
        \pmatrix{ 4 + 4 \cr 10  -2 \cr }
     \quad = \quad 
        \pmatrix{ 8 \cr 8 \cr }
$$


$$
{\bf x } = \pmatrix{ 2 \cr 2 \cr } \quad {\rm vs\ } \quad 
    {\bf A x } = \pmatrix{ 8 \cr 8 \cr }
$$

$$
{\bf A x } \quad = \quad \lambda \thinspace {\bf x}
$$


$$
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ x \cr y \cr } 
     \quad = \quad 
        \lambda \thinspace \pmatrix{ x \cr y \cr }
$$

$$
\eqalign{
2 x \ +\  2 y \quad &= \quad \lambda x \cr
5 x \ -\  1 y \quad &= \quad \lambda y \cr
}
$$

$$
2 y \quad = \quad (\lambda - 2) x \quad \quad \Rightarrow  \quad \quad 
      y \quad = \quad {{1}\over{2}} (\lambda - 2) x
$$

$$
5x \ - \ {{1}\over{2}} \lambda x \ + \ x \quad = \quad 
          {{1}\over{2}} \lambda^2 x \ - \ \lambda x
$$

$$
{{1}\over{2}} \lambda^2 \ - \ {{1}\over{2}} \lambda \ - \ 6 \quad = \quad 0
$$

$$
\lambda^2 \ - \ \lambda \ -\  12 \quad = \quad 0
$$

$$
\lambda = 4 \quad , \quad \lambda = -3
$$


$$
\eqalign{
2 x \ +\  2 y \quad &= \quad 4 x \cr
5 x \ -\  1 y \quad &= \quad 4 y \cr
}
$$

$$ 
\eqalign{
2x \quad &= \quad 2y \quad \quad \quad \Rightarrow \quad x \ = \ y \cr
5x \quad &= \quad 5y \quad \quad \quad \Rightarrow \quad x \ = \ y \cr
}
$$

$$
{\rm eigenvector\ corresponding\ to\ } \lambda = 4 \quad \rightarrow \quad 
    \pmatrix{ 1 \cr 1 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ 2 \cr 2 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ a \cr a \cr } 
$$


$$
{\rm eigenvector\ corresponding\ to\ } \lambda = -3 \quad \rightarrow \quad 
    \pmatrix{ -2 \cr 5 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ 4 \cr -10 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ -2a \cr 5a \cr } 
$$


$$
{{d^2 x_1}\over{dt^2}} \ + \ 2 {{k}\over{m}} x_1 \ - \ {{k}\over{m}} x_2 \quad = \quad 0 
$$

$$
{{d^2 x_2}\over{dt^2}} \ - \   {{k}\over{m}} x_1 \ + \ 2 {{k}\over{m}} x_2 \quad = \quad 0 
$$


$$
-2 {{k}\over{m}} x_1 \ + \ {{k}\over{m}} x_2 \quad = \quad {{d^2 x_1}\over{dt^2}} 
$$

$$
 {{k}\over{m}} x_1 \ - \ 2 {{k}\over{m}} x_2 \quad = \quad {{d^2 x_2}\over{dt^2}} \ 
$$

$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { x_1 \cr x_2 \cr } 
  \quad = \quad 
%    \pmatrix { {d^2 x_1} / {dt^2} \cr
%               {d^2 x_2} / {dt^2} \cr }
    \pmatrix { {{d^2 x_1}\over{dt^2}} \cr
                 \phantom{x} \cr
               {{d^2 x_2}\over{dt^2}} \cr }
$$

$$
s \quad = \quad \pmatrix { a x_1 \cr b x_2 }
$$

$$
{{d^2 s}\over{dt^2}} \quad = \quad  - \omega^2 s
$$

$$
{{d^2 x_1}\over{dt^2}} \quad = \quad  - \omega^2 x_1 
$$

$$
{{d^2 x_2}\over{dt^2}} \quad = \quad  - \omega^2 x_2 
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a x_1 \cr b x_2 \cr } 
  \quad = \quad 
%    \pmatrix { {d^2 x_1} / {dt^2} \cr
%               {d^2 x_2} / {dt^2} \cr }
    - \omega^2 \pmatrix { a x_1  \cr
                          b x_2  \cr }
$$


$$
{\bf A} {\bf s} \quad = \quad \lambda \thinspace {\bf s}
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
    - \omega^2 \pmatrix { a   \cr
                          b  \cr }
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad 
    + \quad \omega^2 \pmatrix { a   \cr
                          b  \cr }
  \quad = \quad 0
$$

$$
- 2 ({{k}\over{m}}) a \ + \ 1 ({{k}\over{m}}) b \ + \ \omega^2 a \quad = \quad 0
$$

$$
  1 ({{k}\over{m}}) a \ - \ 2 ({{k}\over{m}}) b \ + \ \omega^2 b \quad = \quad 0
$$

$$
b \quad = \quad \left(  {{\omega^2 - 2{{k}\over{m}} } \over{- {{k}\over{m}} } } \right) a
$$

$$
1 ({{k}\over{m}}) a  \ + \
    \left( -2 {{k}\over{m}} + \omega^2 \right) b \quad = \quad 0
$$

$$
1 ({{k}\over{m}}) a  \ + \
    \left( -2 {{k}\over{m}} + \omega^2 \right) 
      \left(  {{\omega^2 - 2{{k}\over{m}} } \over{- {{k}\over{m}} } } \right) a
 \quad = \quad 0
$$

$$
-1 \left({{k}\over{m}}\right)^2 \ + \ 
   4 \left({{k}\over{m}}\right)^2 \ - \ 
    4 \left({{k}\over{m}}\right) \omega^2 \ + \
   \omega^4 \quad = \quad 0
$$

$$
\lambda \quad = \quad \omega^2
$$

$$
3 \left({{k}\over{m}}\right)^2 \ - \ 
    4 \left({{k}\over{m}}\right) \lambda \ + \
   \lambda^2 \quad = \quad 0
$$


$$
\lambda \quad = \quad \omega^2 \quad = \quad 
     1 {{k}\over{m}} \quad {\rm or \ } \quad 3 {{k}\over{m}} 
$$

$$
\eqalign{
\omega_1 \quad &= \quad \sqrt{ {{k}\over{m}} } \cr
\omega_2 \quad &= \quad \sqrt{ {{3k}\over{m}} } \cr
}
$$

$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad 
    + \quad 1 ({{k}\over{m}}) \pmatrix { a   \cr
                          b  \cr }
  \quad = \quad 0
$$

$$
- 2 ({{k}\over{m}}) a \ + \ 1 ({{k}\over{m}}) b \ + \ 1 ({{k}\over{m}}) a \quad = \quad 0
$$

$$
 1 ({{k}\over{m}}) a \ - \ 2 ({{k}\over{m}}) b \ + \ 1 ({{k}\over{m}}) b \quad = \quad 0
$$

$$
a \quad = \quad b
$$

$$
{\rm eigenvalue\ } \omega_1 \ = \ \sqrt{{k}\over{m}} \  \quad 
   {\rm corresponds\ to\ eigenvector\ } \quad 
   \pmatrix { 1 \cr 1 \cr }
$$

$$
s_1 \quad = \quad 
   \pmatrix { 1 x_1 \cr 1 x_2 \cr } 
      \quad \quad {\rm or } \quad \quad 
s_1 \quad = \quad x_1 \ + \ x_2
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad 
    + \quad 3 ({{k}\over{m}}) \pmatrix { a   \cr
                          b  \cr }
  \quad = \quad 0
$$

$$
- 2 ({{k}\over{m}}) a \ + \ 1 ({{k}\over{m}}) b \ + \ 3 ({{k}\over{m}}) a \quad = \quad 0
$$

$$
 1 ({{k}\over{m}}) a \ - \ 2 ({{k}\over{m}}) b \ + \ 3 ({{k}\over{m}}) b \quad = \quad 0
$$



$$
a \quad = \quad -b
$$

$$
{\rm eigenvalue\ } \omega_2 \ = \ \sqrt{{3k}\over{m}} \  \quad 
   {\rm corresponds\ to\ eigenvector\ } \quad 
   \pmatrix { 1 \cr -1 \cr }
$$


$$
s_2 \quad = \quad 
   \pmatrix { 1 x_1 \cr -1 x_2 \cr } 
      \quad \quad {\rm or } \quad \quad 
s_2 \ = \ x_1 \ - \ x_2 
      \quad {\rm or\ } \quad 
s_2 \ = \ x_2 \ - \ x_1 
$$



$$
\left(  k/m  \right) 
  \pmatrix{ -2  &  1 \cr 
             1 &  -2 \cr }  
   \pmatrix { a \cr b \cr } 
  \quad 
    - \quad \lambda \pmatrix { a   \cr
                               b  \cr }
  \quad = \quad 0
$$

$$
\left(  k/m  \right) 
  \pmatrix{ -2 - \lambda &  1 \cr 
             1 &  -2 - \lambda \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 0
$$

$$
% \eqalign{
%  {\rm determinant } \ 
%   \pmatrix{ -2 - \lambda &  1 \cr 
%              1 &  -2 - \lambda \cr }  
%  \quad &= \quad 
%     (-2 - \lambda)(-2 - \lambda) - (1)(1) 
%  } \cr
%      &= \quad (4 \ + 4 \lambda \ +  )
% }
$$

$$
F_1 \quad = \quad -k x_1 \ + \ 2k(x_2 - x_1) 
$$

$$
F_2 \quad = \quad -2k(x_2 - x_1) \ - \ 3k x_2 
$$


$$
\eqalign{
  F_1 \quad &= \quad -3k x_1 \ + \ 2k x_2  \cr
  F_2 \quad &= \quad \phantom{-} 2k x_1 \ - \ 5k x_2  \cr
}
$$


$$
\eqalign{
  m \thinspace {{d^2 x_1}\over{dt^2}} 
     \quad &= \quad -3k x_1 \ + \ 2k x_2  \cr
  m \thinspace {{d^2 x_2}\over{dt^2}} 
     \quad &= \quad \phantom{-} 2k x_1 \ - \ 5k x_2  \cr
}
$$


$$
\eqalign{
   \left( {{k}\over{m}} \right)
        \bigl[  -3 x_1 \ + \ 2 x_2  \bigr] \
     \quad &= \quad 
   \thinspace {{d^2 x_1}\over{dt^2}}  \cr
   \left( {{k}\over{m}} \right)
        \bigl[2 x_1 \ - \ 5 x_2 \bigr]   
     \quad &= \quad 
   \thinspace {{d^2 x_2}\over{dt^2}}  \cr
}
$$

$$
\left( {{k}\over{m}} \right) \thinspace
  \pmatrix{ -3 & 2 \cr 
             2 & -5 \cr }  
   \pmatrix { x_1 \cr x_2 \cr } 
  \quad = \quad 
%    \pmatrix { {d^2 x_1} / {dt^2} \cr
%               {d^2 x_2} / {dt^2} \cr }
    \pmatrix { {{d^2 x_1}\over{dt^2}} \cr
                 \phantom{x} \cr
               {{d^2 x_2}\over{dt^2}} \cr }
$$


$$
s \quad = \quad a x_1 \ + \ b x_2
$$

$$
{{d^2 s}\over{dt^2}} \quad = \quad
   - \omega^2 \thinspace s
$$


$$
\left( {{k}\over{m}} \right) \thinspace
  \pmatrix{ -3 & 2 \cr 
             2 & -5 \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
    - \omega^2 \thinspace
   \pmatrix { a \cr b \cr } 
$$


$$
\eqalign{
  -3 a \ + \ 2 b 
     \quad &= \quad 
           - \left( {{\omega^2}\over{k/m}}\right) \thinspace a \cr
  2 a \ - \ 5 b 
     \quad &= \quad 
    - \left( {{\omega^2}\over{k/m}} \right)  \thinspace b \cr
}
$$

$$
b \quad = \quad -{{1}\over{2}} {{\omega^2}\over{k/m}} \thinspace a
             \ + \ {{3}\over{2}} \thinspace a
$$

$$
{{\omega^4}\over{{k^2/m^2}}} \thinspace 
          \ + \ 
          8 {{\omega^2}\over{k/m}} 
          \ + \ 
          11
   \quad = \quad 
          0
$$

$$
\left( {{k}\over{m}} \right) \thinspace
  \pmatrix{ -3 & 2 \cr 
             2 & -5 \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
    \lambda \thinspace
   \pmatrix { a \cr b \cr } 
$$

$$
  \pmatrix{ -3 & 2 \cr 
             2 & -5 \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
    \lambda \thinspace
   \pmatrix { a \cr b \cr } 
$$

$$
  \pmatrix{ -3 & 2 \cr 
             2 & -5 \cr }  
   \pmatrix { a \cr b \cr } 
   \ - \ 
    \lambda \thinspace
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
      0
$$


$$
  \pmatrix{ -3 - \lambda & 2 \cr 
             2 & -5 - \lambda \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
      0
$$

$$
(-3 - \lambda) (-5 - \lambda) \ - \ (2) (2) 
$$

$$
\lambda^2 \ + \ 8 \lambda \ + \ 11
   \quad = \quad 
   0
$$

$$
\lambda \quad = \quad - \omega^2
$$

$$
\eqalign{
\omega_1 \quad &= \quad \sqrt{ 1.76 {{k}\over{m}} } \cr
\omega_2 \quad &= \quad \sqrt{ 6.24 {{k}\over{m}} } \cr
}
$$


$$
\left( {{k}\over{m}} \right) \thinspace
  \pmatrix{ -3 & 2 \cr 
             2 & -5 \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
    - 1.76 {{k}\over{m}} \thinspace
   \pmatrix { a \cr b \cr } 
$$


$$
\left( {{k}\over{m}} \right) \thinspace
  \pmatrix{ -3 & 2 \cr 
             2 & -5 \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
    - 6.24 {{k}\over{m}} \thinspace
   \pmatrix { a \cr b \cr } 
$$

$$
\eqalign{
{\omega_1}^2 &= 1.76 {{k}\over{m}} \quad \rightarrow \quad
      a \quad = \quad 1.61 \thinspace b \cr
{\omega_2}^2 &= 6.24 {{k}\over{m}} \quad \rightarrow \quad
      a \quad = \quad -0.62 \thinspace b \cr
}
$$

$$
\eqalign{
s_1 \quad = \quad {\phantom{-}}1.61 x_1  \ + \ x_2 \cr
s_2 \quad = \quad -0.62 x_1  \ + \ x_2 \cr
}
$$

$$
\eqalign{
s_1(t) \quad &= \quad A_1 \cos(\omega_1 t \ + \ \phi_1) \cr
s_2(t) \quad &= \quad A_2 \cos(\omega_2 t \ + \ \phi_2) \cr
}
$$


$$
\eqalign{
s_1(t) \quad &= \quad L \cos(\omega_1 t ) \cr
s_2(t) \quad &= \quad L \cos(\omega_2 t ) \cr
}
$$



$$
\lambda \quad \equiv \quad - \omega^2 \thinspace ({{m}\over{k}})
$$

$$
\omega \quad \equiv \quad \sqrt{ -\lambda \thinspace {{k}\over{m}} }
$$




\bye
