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

F_{1} = −2kx_{1} + kx_{2}

F_{2} = kx_{1} − 2kx_{2} + kx_{3}

F_{3} = kx_{2} − 2kx_{3}

(\frac{k}{m}) ( \begin{matrix}−2 & 1 & 0 \\ 1 & −2 & 1 \\ 0 & 1 & −2 \end{matrix} ) ( \begin{matrix}ax_{1} \\ bx_{2} \\ cx_{3} \end{matrix} ) = −ω^{2} ( \begin{matrix}ax_{1} \\ bx_{2} \\ cx_{3} \end{matrix} )

\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{aligned}ω_{1} & = \sqrt{\frac{k}{m}} \\ ω_{2} & = \sqrt{\frac{3k}{m}} \end{aligned}

( \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_{1} \\ 1x_{2} \end{matrix} )  or  s_{1} = x_{1} + x_{2}

( \begin{matrix}−2\frac{k}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) + 3(\frac{k}{m}) ( \begin{matrix}a \\ b \end{matrix} ) = 0

−2(\frac{k}{m})a + 1(\frac{k}{m})b + 3(\frac{k}{m})a = 0

1(\frac{k}{m})a − 2(\frac{k}{m})b + 3(\frac{k}{m})b = 0

a = −b

eigenvalue ω_{2} = \sqrt{\frac{3k}{m}} corresponds to eigenvector ( \begin{matrix}1 \\ −1 \end{matrix} )

s_{2} = ( \begin{matrix}1x_{1} \\ −1x_{2} \end{matrix} )  or  s_{2} = x_{1} − x_{2} or s_{2} = x_{2} − x_{1}

−ω_{1}^{2} = (\sqrt{2} − 2)\frac{k}{m}  ⇒  ω_{1} = \sqrt{(2 − \sqrt{2})\frac{k}{m}} = 0.765 \sqrt{\frac{k}{m}}

−ω_{2}^{2} = −2\frac{k}{m}  ⇒  ω_{2} = \sqrt{2\frac{k}{m}} = 1.414 \sqrt{\frac{k}{m}}

−ω_{3}^{2} = (−\sqrt{2} − 2)\frac{k}{m}  ⇒  ω_{3} = \sqrt{(2 + \sqrt{2})\frac{k}{m}} = 1.848 \sqrt{\frac{k}{m}}

s_{1} = ( \begin{matrix}1 \\ √ & \overline{2} \\ 1 \end{matrix} ) = x_{1} + \sqrt{2}x_{2} + x_{3}

s_{2} = ( \begin{matrix}−1 \\ 0 \\ 1 \end{matrix} ) = −x_{1} + 0x_{2} + x_{3}

s_{3} = ( \begin{matrix}1 \\ −\sqrt{2} \\ 1 \end{matrix} ) = x_{1} − \sqrt{2}x_{2} + x_{3}

s_{1}(t) = A_{1} cos(ω_{1}t + ϕ_{1})

s_{2}(t) = A_{2} cos(ω_{2}t + ϕ_{2})

s_{3}(t) = A_{3} cos(ω_{3}t + ϕ_{3})

x_{1} = \frac{1}{4}s_{1} − \frac{1}{2}s_{2} + \frac{1}{4}s_{3}

s_{1} = ( \begin{matrix}1 \\ 1 \end{matrix} )

s_{2} = ( \begin{matrix}1 \\ −1 \end{matrix} )

s_{1} = ( \begin{matrix}1 \\ 1 \end{matrix} )

s_{2} = ( \begin{matrix}1 \\ −1 \end{matrix} )

(\frac{k}{m}) ( \begin{matrix}−2 & 1 & 0 & 0 \\ 1 & −2 & 1 & 0 \\ 0 & 1 & −2 & 1 \\ 0 & 0 & 1 & −2 \end{matrix} ) ( \begin{matrix}ax_{1} \\ bx_{2} \\ cx_{3} \\ dx_{4} \end{matrix} ) = −ω^{2} ( \begin{matrix}ax_{1} \\ bx_{2} \\ cx_{3} \\ dx_{4} \end{matrix} )

ω_{1}^{2} = 0.38197\frac{k}{m}  ⇒  ω_{1} = 0.618\sqrt{\frac{k}{m}}

ω_{2}^{2} = 3.61803\frac{k}{m}  ⇒  ω_{2} = 1.902\sqrt{\frac{k}{m}}

ω_{3}^{2} = 2.61803\frac{k}{m}  ⇒  ω_{3} = 1.618\sqrt{\frac{k}{m}}

ω_{4}^{2} = 1.38197\frac{k}{m}  ⇒  ω_{4} = 1.176\sqrt{\frac{k}{m}}

s_{1} = ( \begin{matrix}1 \\ 1.61803 \\ 1.61803 \\ 1 \end{matrix} )

s_{2} = ( \begin{matrix}−1 \\ −0.61803 \\ 0.61803 \\ 1 \end{matrix} )

s_{3} = ( \begin{matrix}1 \\ −0.61803 \\ −0.61803 \\ 1 \end{matrix} )

s_{4} = ( \begin{matrix}−1 \\ 1.61803 \\ −1.61803 \\ 1 \end{matrix} )

ω_{n} = 2 (\sqrt{\frac{k}{m}}) sin [\frac{nπ}{2(N + 1)}] = 2 ω_{0} sin [\frac{nπ}{2(N + 1)}]

n << N

[\frac{nπ}{2(N + 1)}] << 1

ω_{n} ≃ 2 (\sqrt{\frac{k}{m}}) [\frac{nπ}{2(N + 1)}] ≃ 2 ω_{0} [\frac{nπ}{2(N + 1)}]

ω_{n} ≃ (2 ω_{0} [\frac{π}{2(N + 1)}]) n

ω_{1} ≃ 2 ω_{0} [\frac{π}{2(N + 1)}]

ω_{n} ≃ n ω_{1}

\frac{F}{ΔL} = \frac{A}{L} Y

\frac{F}{ΔL} = \frac{A}{L} Y \frac{F}{Δx} = k

ω_{0} = \sqrt{\frac{k}{m}}

ω_{0} = \sqrt{\frac{AY/L}{ρAL}}

f_{0} = \frac{ω_{0}}{2π} = 2.4 × 10^{12}Hz

λ_{0} = \frac{c}{f_{0}} = 122 μm

ω_{N} = 2ω_{0}  ⇒  λ_{N} = \frac{1}{2} λ_{0}