matrix A = ( \begin{matrix}2 & 2 \\ 5 & −1 \end{matrix} )

vector x = ( \begin{matrix}1 \\ 2 \end{matrix} )

Ax = ( \begin{matrix}2 & 2 \\ 5 & −1 \end{matrix} ) ( \begin{matrix}1 \\ 2 \end{matrix} ) = ( \begin{matrix}2 + 4 \\ 5 − 2 \end{matrix} ) = ( \begin{matrix}6 \\ 3 \end{matrix} )

x = ( \begin{matrix}1 \\ 2 \end{matrix} ) vs Ax = ( \begin{matrix}6 \\ 3 \end{matrix} )

vector x = ( \begin{matrix}1 \\ 1 \end{matrix} )

Ax = ( \begin{matrix}2 & 2 \\ 5 & −1 \end{matrix} ) ( \begin{matrix}1 \\ 1 \end{matrix} ) = ( \begin{matrix}2 + 2 \\ 5 − 1 \end{matrix} ) = ( \begin{matrix}4 \\ 4 \end{matrix} )

x = ( \begin{matrix}1 \\ 1 \end{matrix} ) vs Ax = ( \begin{matrix}4 \\ 4 \end{matrix} )

Ax = ( \begin{matrix}4 \\ 4 \end{matrix} ) = 4x

vector x = ( \begin{matrix}2 \\ 2 \end{matrix} )

Ax = ( \begin{matrix}2 & 2 \\ 5 & −1 \end{matrix} ) ( \begin{matrix}2 \\ 2 \end{matrix} ) = ( \begin{matrix}4 + 4 \\ 10 − 2 \end{matrix} ) = ( \begin{matrix}8 \\ 8 \end{matrix} )

x = ( \begin{matrix}2 \\ 2 \end{matrix} ) vs Ax = ( \begin{matrix}8 \\ 8 \end{matrix} )

Ax = λ x

( \begin{matrix}2 & 2 \\ 5 & −1 \end{matrix} ) ( \begin{matrix}x \\ y \end{matrix} ) = λ ( \begin{matrix}x \\ y \end{matrix} )

\begin{aligned}2x + 2y & = λx \\ 5x − 1y & = λy \end{aligned}

2y = (λ − 2)x  ⇒  y = \frac{1}{2}(λ − 2)x

5x − \frac{1}{2}λx + x = \frac{1}{2}λ^{2}x − λx

\frac{1}{2}λ^{2} − \frac{1}{2}λ − 6 = 0

λ^{2} − λ − 12 = 0

λ = 4 , λ = −3

\begin{aligned}2x + 2y & = 4x \\ 5x − 1y & = 4y \end{aligned}

\begin{aligned}2x & = 2y  ⇒ x = y \\ 5x & = 5y  ⇒ x = y \end{aligned}

eigenvector corresponding to λ = 4 → ( \begin{matrix}1 \\ 1 \end{matrix} ) or ( \begin{matrix}2 \\ 2 \end{matrix} ) or ( \begin{matrix}a \\ a \end{matrix} )

eigenvector corresponding to λ = −3 → ( \begin{matrix}−2 \\ 5 \end{matrix} ) or ( \begin{matrix}4 \\ −10 \end{matrix} ) or ( \begin{matrix}−2a \\ 5a \end{matrix} )

\frac{d^{2}x_{1}}{dt^{2}} + 2\frac{k}{m}x_{1} − \frac{k}{m}x_{2} = 0

\frac{d^{2}x_{2}}{dt^{2}} − \frac{k}{m}x_{1} + 2\frac{k}{m}x_{2} = 0

−2\frac{k}{m}x_{1} + \frac{k}{m}x_{2} = \frac{d^{2}x_{1}}{dt^{2}}

\frac{k}{m}x_{1} − 2\frac{k}{m}x_{2} = \frac{d^{2}x_{2}}{dt^{2}}

( \begin{matrix}−2\frac{k}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) = ( \frac{d^{2}x_{1}}{dt^{2}} ; \frac{d^{2}x_{2}}{dt^{2}} )

s = ( \begin{matrix}ax_{1} \\ bx_{2} \end{matrix} )

\frac{d^{2}s}{dt^{2}} = −ω^{2}s

\frac{d^{2}x_{1}}{dt^{2}} = −ω^{2}x_{1}

\frac{d^{2}x_{2}}{dt^{2}} = −ω^{2}x_{2}

( \begin{matrix}−2\frac{k}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} \end{matrix} ) ( \begin{matrix}ax_{1} \\ bx_{2} \end{matrix} ) = −ω^{2} ( \begin{matrix}ax_{1} \\ bx_{2} \end{matrix} )

As = λ s

( \begin{matrix}−2\frac{k}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) = −ω^{2} ( \begin{matrix}a \\ b \end{matrix} )

( \begin{matrix}−2\frac{k}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) + ω^{2} ( \begin{matrix}a \\ b \end{matrix} ) = 0

−2(\frac{k}{m})a + 1(\frac{k}{m})b + ω^{2}a = 0

1(\frac{k}{m})a − 2(\frac{k}{m})b + ω^{2}b = 0

b = (\frac{ω^{2} − 2\frac{k}{m}}{−\frac{k}{m}} ) a

1(\frac{k}{m})a + (−2\frac{k}{m} + ω^{2}) b = 0

1(\frac{k}{m})a + (−2\frac{k}{m} + ω^{2}) (\frac{ω^{2} − 2\frac{k}{m}}{−\frac{k}{m}} ) a = 0

−1 (\frac{k}{m})^{2} + 4 (\frac{k}{m})^{2} − 4 (\frac{k}{m}) ω^{2} + ω^{4} = 0

λ = ω^{2}

3 (\frac{k}{m})^{2} − 4 (\frac{k}{m}) λ + λ^{2} = 0

λ = ω^{2} = 1\frac{k}{m} or 3\frac{k}{m}

( \begin{matrix}−2\frac{k}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) + 1(\frac{k}{m}) ( \begin{matrix}a \\ b \end{matrix} ) = 0

−2(\frac{k}{m})a + 1(\frac{k}{m})b + 1(\frac{k}{m})a = 0

1(\frac{k}{m})a − 2(\frac{k}{m})b + 1(\frac{k}{m})b = 0

a = b

eigenvalue ω_{1} = \sqrt{\frac{k}{m}} corresponds to eigenvector ( \begin{matrix}1 \\ 1 \end{matrix} )

s_{1} = ( \begin{matrix}1x \\ 1y \end{matrix} )  or  s_{1} = x_{1} + x_{2}