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Eigen Decomposition

๐Ÿ™…โ€โ™‚๏ธํœด๋Œ€ํฐ์œผ๋กœ ๋ณผ ๋•Œ ํ˜น์‹œ ๊ธ€์ž๋‚˜ ์ˆซ์ž๊ฐ€ ํ™”๋ฉด์— ๋‹ค ์•ˆ๋‚˜์˜ค๋ฉด, ํœด๋Œ€ํฐ ๊ฐ€๋กœ๋กœ ๋Œ๋ฆฌ์‹œ๋ฉด ๋ฉ๋‹ˆ๋‹ค

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0. ๋“ค์–ด๊ฐ€๋ฉฐ

1. Eigen decomposition (long provement)
 1-1. Eigen decomposition to diagonalization
 1-2. Eigen decomposition ์žฅ์ (5๊ฐœ) & ๊ฟ€ํŒ(3๊ฐœ)

2. feature of symmetric matrix
 2-1. ํ‘œํ˜„
 2-2. ์‹ค์ƒํ™œ ์‘์šฉ
 2-3. ์ƒˆ๋กœ์šด ํ•ด์„  

3. ๋Œ€๊ฐํ™” ํŒ๋ณ„๋ฒ•
 โ˜…๋Œ€๊ฐํ™” ๊ฐ€๋Šฅ, ๋ถˆ๊ฐ€๋Šฅ ์‚ฌ๋ก€
 3-1 ์ค‘๋ณต๋„  
 3-2 ๋‹ฎ์Œ ๋ถˆ๋ณ€๋Ÿ‰

4. ์ผ€์ผ๋ฆฌ-ํ•ด๋ฐ€ํ„ด ์ •๋ฆฌ

5. ์—ฐ์Šต๋ฌธ์ œ (3๊ฐœ) ๊ณ ์œ ๊ฐ’,๊ณ ์œ ๋ฒกํ„ฐ 2๊ฐœ, ์ผ€์ผ๋ฆฌํ•ด๋ฐ€ํ„ด 1๊ฐœ

0. ๋“ค์–ด๊ฐ€๋ฉฐ

์šฐ์„  ์š”์•ฝ๋ณธ์ธ๋ฐ ์ž ์‹œ ํ›‘์–ด๋ณด๊ณ  ์ง€๋‚˜๊ฐ€๋Š” ๊ฒƒ๋„ ๋‚˜์˜์ง„ ์•Š์„ ๊ฒƒ ๊ฐ™์Šต๋‹ˆ๋‹ค.
EVD ์ฆ‰ ๊ณ ์œณ๊ฐ’ ๋ถ„ํ•ด๋ฅผ ์œ„ํ•œ ์‹์€ ์•„๋ž˜์™€ ๊ฐ™์ด ๋‚˜ํƒ€๋‚ผ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค.
ํ–‰๋ ฌAx๊ฐ€ ์žˆ์„ ๋•Œ
Ax=ฮป1q1qT1x+ฮป2q2qT2x+โ‹ฏ+ฮปnqnqTnx

์—ฌ๊ธฐ์„œ dim reduction์„ ํ•˜์—ฌ ๊ณ ์œ ๋ฒกํ„ฐ 2๊ฐœ๋งŒ ์‚ฌ์šฉํ• ์‹œ
Axโ‰ˆฮป1q1qT1x+ฮป2q2qT2x ์•„๋ฌดํŠผ ์„œ๋ก ์ด ์ข€ ๊ธธ์—ˆ๋Š”๋ฐ ๊ฑฐ๋‘์ ˆ๋ฏธํ•˜๊ณ ,

์ด๋ฒˆ ๊ธ€์—์„œ์˜ ์ฃผ์š” ๋‚ด์šฉ์ธ ํ–‰๋ ฌ์„ ๊ณ ์œณ๊ฐ’๊ณผ ๊ณ ์œ ๋ฒกํ„ฐ๋กœ ๋ถ„ํ•ดํ•˜๋Š” ๊ณผ์ •์ธ Eigenvalue Decomposition(๊ณ ์œณ๊ฐ’ ๋ถ„ํ•ด)์— ๋Œ€ํ•œ ๋‚ด์šฉ๊ณผ, symmetric matrix์˜ ์‹ค์ƒํ™œ ์‘์šฉ ๋ฐ ์ƒˆ๋กœ์šด ํ•ด์„์— ๊ด€ํ•ด ์•Œ์•„๋ณผ ๊ฒƒ์ž…๋‹ˆ๋‹ค


1. Eigen decomposition (long provement)

โ€”โ€”โ€”โ€”โ€”โ€”โ€” ๋œป โ€”โ€”โ€”โ€”โ€”โ€”โ€”-
characteristic equation(๊ณ ์œ ๋ฐฉ์ •์‹)
det(ฮณInโˆ’M) โ‡’ ๊ณ ์œ ๊ฐ’, ๊ณ ์œ ๋ฒกํ„ฐ๋ฅผ ์ฐพ๋Š” ๊ณผ์ •

diagonalization
A=VฮณVโˆ’1
โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-





1-1. Eigen decomposition to diagonalization

A2โˆ—2 ๊ฐ€ ์žˆ๋‹ค๊ณ  ํ•˜์ž ์ด๋•Œ
eigen value (2๊ฐœ)   ฮณ1,ฮณ2
eigen vector(๋ฌด์กฐ๊ฑด independant) 2๊ฐœ   v1, v2

๊ทธ๋Ÿฌ๋ฉด ์ž์—ฐ์Šค๋Ÿฝ๊ฒŒ Av1=ฮณ1v1,       Av2=ฮณ2v2 ๊ฐ€ ๋œ๋‹ค
์—ฌ๊ธฐ์„œ 2๊ฐœ์˜ ์ˆ˜์‹์„ ํ•˜๋‚˜๋กœ ํ•ฉ์ณ๋ณด์ž
A[v1,   v2]=[ฮณ1v1,   ฮณ2v2]

์˜ค ์ด์‹์„ ์ด๋ ‡๊ฒŒ ๋ฐ”๊ฟ€ ์ˆ˜๋„ ์žˆ๊ตฐ
=[v1,  v2][ฮณ100ฮณ2]

์ž ์ด์‹์—์„œ [v1,   v2]๋Š” v๋งŒ ๋ชจ์•„๋†“์€ ๊ฒƒ์ด๋‹ˆ ํ–‰๋ ฌV๋ผ ํ•˜์ž
V=[v1,   v2], ๋งˆ์ฐฌ๊ฐ€์ง€๋กœ ฮณ=[ฮณ100ฮณ2]
์–ด ๊ทธ๋Ÿฌ๋ฉด ์‹์„ ์ด๋ ‡๊ฒŒ๋„ ๋ฐ”๊ฟ€ ์ˆ˜ ์žˆ๊ฒ ๋‹ค โ‡’AV=Vฮณ

๊ทธ๋Ÿด ๋•Œ, v1,   v2๋Š” independantํ•œ vector๋กœ ์‚ผ์œผ๋‹ˆ ๊ทธ๋Ÿผ ์ด๊ฑด ์›๋ž˜ rank๊ฐ€ 2 by 2์ธ ํ–‰๋ ฌ์ธ๋ฐ๋„ 2๊ฐœ๋‹ค
์ฆ‰ invertableํ•˜๋‹ค   why? (detโ‰ 0 ์ด๋‹ˆ๊นŒ ์—ญํ–‰๋ ฌ ์กด์žฌํ•ด์„œ)

๊ทธ๋ž˜์„œ ์ด๋ ‡๊ฒŒ ์‹์„ ๋ฐ”๊ฟ€ ์ˆ˜ ์žˆ๋‹ค โ‡’A=VฮณVโˆ’1
๐Ÿค”๋งŒ์•ฝ์— ์—ฌ๊ธฐ์„œ ์‹์„ ์ด๋ ‡๊ฒŒ ๋ฐ”๊พธ๋ฉด? โ‡’Vโˆ’1AV=ฮณ
๋‹น์—ฐํžˆ! ฮณ๋Š” diagonal matrix๋‹ˆ๊นŒ eigen decomposition์ด ๋˜๋Š” A๋ฅผ โ€œdiagonalizableํ•˜๋‹คโ€๋ผ๊ณ  ํ•œ๋‹ค

๋‹ค์‹œ ๋งํ•ด
Anโˆ—nโ‡’diagonalizable โ‡” independant Eigen vector๊ฐ€ n๊ฐœ๋‹ค
โ€”โ€”โ€”โ€”โ€”โ€”โ€” ๋œป โ€”โ€”โ€”โ€”โ€”โ€”โ€”-
n by n์˜ Aํ–‰๋ ฌ์ด diagonalizableํ•˜๋ฉด independant Eigen vector๊ฐ€ n๊ฐœ๋ผ๋Š” ๊ฒƒ๊ณผ ๋™์น˜๋‹ค
โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”โ€”-


1-2. Eigen decomposition ์žฅ์ (5๊ฐœ) & ๊ฟ€ํŒ(3๊ฐœ)

๐Ÿคตโ€โ™€๏ธ์žฅ์ 

(1) Ak    ex) A3=VฮณVโˆ’1 โ‹… Vโˆ’1ฮณV โ‹… VฮณVโˆ’1 = Vฮณ3Vโˆ’1

----------------sol-(1)-------------------
ฮณk๋Š”[ฮณk00ฮณk]   ์ธ๋ฐ ์ œ๊ณฑ, ์„ธ์ œ๊ณฑ, ๋„ค์ œ๊ณฑโ€ฆ์€ ฮณ ๊ฐ’๋งŒ ๋ฐ”๊ฟ”์ฃผ๋ฉด ๋˜์–ด ๊ณ„์‚ฐ์ด ํŽธํ•˜๋‹ค
-------------------------------------------

(2) Aโˆ’1= (VฮณVโˆ’1)โˆ’1=Vฮณโˆ’1Vโˆ’1

----------------sol-(2)-------------------
[ฮณ100ฮณ2]โˆ’1=[1ฮณ1001ฮณ2]
์ž ์—ฌ๊ธฐ์„œ Aโˆ’1A ํ™•์ธํ•˜์ž โ‡’ Vฮณโˆ’1Vโˆ’1 โ‹… VฮณVโˆ’1 ํ•˜๋ฉด ๋ฐ”๋กœ ํ•ญ๋“ฑํ–‰๋ ฌI๊ฐ€ ๋˜๋„ค
๋งˆ์ฐฌ๊ฐ€์ง€๋กœ ์œ„์น˜๋ฅผ ๋ฐ”๊ฟ” AAโˆ’1 ํ•ด๋„ ํ•ญ๋“ฑํ–‰๋ ฌI ๋‚˜์˜ด
-------------------------------------------

(3) det(A)= det(VฮณVโˆ’1)=det(V)det(ฮณ)det(Vโˆ’1)โ‡’det(ฮณ)=ฮณ1โ‹…ฮณ2โ‹ฏ=โˆni=1ฮณi


(4) tr(A)=(VฮณVโˆ’1)

----------------sol-(4)-------------------
์ฐธ๊ณ :
trace: ์ •๋ฐฉํ–‰๋ ฌ์˜ ๋Œ€๊ฐ์„ฑ๋ถ„์˜ ํ•ฉ ex)   A=[1004]tr(A)=1+4=5
๊ทธ๋ฆฌ๊ณ  tr(ABC)= tr(BCA)=tr(CBA)
์ž ์ด๊ฒŒ ์œ„์น˜๋ฅผ ๋ฐ”๊ฟ”๋„ ๊ฐ™๋‹ค๋Š” ์„ฑ์งˆ์„ ์ด์šฉํ•˜์—ฌ

tr(VฮณVโˆ’1)=tr(ฮณVโˆ’1V)=tr(ฮณ)
โ‡’ฮณ1+ฮณ2+โ‹ฏโˆด
-------------------------------------------

(5) rank-difficient \color{red}{\Leftrightarrow} det(A)=0 \color{red}{\Leftrightarrow} 0 ์ธ eigen value๊ฐ€ 1๊ฐœ ์ด์ƒ ์กด์žฌ

----------------ํ•ด์„-(5)-------------------
rank-dificient๋Š” det=0์ธ ๊ฒƒ๊ณผ ๋™์น˜์ธ๋ฐ det๋Š” \gamma ๋ฅผ ์‹น ๊ณฑํ•œ ๊ฒƒ์ด๋‹ˆ ๊ทธ๋ง์€ ์ฆ‰์Šจ
0์ธ eigen value๊ฐ€ 1๊ฐœ ์ด์ƒ ์กด์žฌํ•œ๋‹ค๋Š” ๋œป
-------------------------------------------


๐Ÿฏ๊ฟ€ํŒ

(1) A^T์˜ Eigen value \color{red}{=} A์˜ Eigen value

----------------sol-(1)-------------------
why? \Rightarrow det(A-\gamma I) = det(A-\gamma I)^{T}
์™œ๋ƒํ•˜๋ฉด det(A) = det(A)^T ๋ผ์„œ
\therefore det(A-\gamma I) = det(A^T-\gamma I)
-------------------------------------------

(2) A๊ฐ€ orghogonal matrix๋ฉด   \gamma_i = \pm 1 ์ด๋‹ค

----------------sol-(2)-------------------
์šฐ์„  orthogonal matrix๋ฅผ Q๋กœ ๋‘์ž ๋นจ๋นจ๋นจ๋นจ๊ฐ•, ๊ทธ๋ฆฌ๊ณ  ๊ฑฐ๊ธฐ์— ํ–‰๋ ฌ v๋ฅผ ํ†ต๊ณผ์‹œํ‚ค์ž
QV=\gamma V
\left(Qv\right)^{T}Qv= V^TQ^TQV=V^TV= \parallel V \parallel_2^2
์–ด?   ์—ฌ๊ธฐ์„œ \left(\gamma V\right)^{T}\gamma V ์ด๋ ‡๊ฒŒ ๊ณ ์น  ์ˆ˜๋„ ์žˆ๋Š”๋ฐ    ๊ทธ๋Ÿฌ๋ฉด \color{red}{\therefore} \gamma^2\parallel V \parallel_2^2
-------------------------------------------

(3) A๊ฐ€ positive semi-difinite(P.S.D)๋ฉด \color{red}{\Leftrightarrow} \quad \gamma_i \ge 0

----------------sol-(3)-------------------
(์ด๋•Œ A= A^T\color{red}{,} \quad ์ฆ‰ symmetric matrix)

๊ทธ๋‚˜์ €๋‚˜ PSD??? โฌ‡๏ธ
z^TAz \ge 0 ์ด๊ณ  ์ด๊ฒƒ์€ z๋ฅผ ์„ ํ˜•๋ณ€ํ™˜ํ•œ ๊ฒƒ์ธ๋ฐ z์™€ ๋‚ด์ ํ–ˆ์„ ๋•Œ ์–‘์ˆ˜๊ฐ€ ๋œ๋‹ค๋Š” ๊ฒƒ์€
์„ ํ˜•๋ณ€ํ™˜์„ ๊ฑฐ์ณ๋„ ์ง๊ตํ•˜๋Š” ํ‰๋ฉด ๋’ค์ชฝ์œผ๋กœ ์•ˆํŠ€์–ด๋‚˜๊ฐ„๋‹ค๋Š” ๋ง์ด๋‹ค
(์ฆ‰ ์–ด๋–ค ๋ฒกํ„ฐ๋ฅผ ํ†ต๊ณผ์‹œํ‚ค๋”๋ผ๋„ ์•„๋ž˜ ๊ทธ๋ฆผ์ฒ˜๋Ÿผ๋งŒ ๋ฐ”๋€๋‹ค)
Desktop View
Desktop View

๋‚ด์ ํ–ˆ์„ ๋•Œ ์Œ์ˆ˜๊ฐ€ ๋˜๋Š” 90^{\circ} ๋ฐฉํ–ฅ์„ ๋„˜์–ด๊ฐ€์ง€ ์•Š๋Š”๋‹ค

์ด์–ด์„œ ์‹์„ ๋ฐ”๊ฟ”๋ณด์ž
Az=\gamma z ๋กœ ๋‘๋ฉด z^T \gamma z \ge 0 ๋œ๋‹ค \color{red}{\Rightarrow} \quad \vert\vert z\vert\vert_2^2 \gamma\ge 0
\color{red}{\therefore} ์ž \vert\vert z\vert\vert_2^2๊ฐ€ ์–‘์ˆ˜๋‹ˆ \gamma๋„ ๋ฌด์กฐ๊ฑด ์–‘์ˆ˜์ด๊ฒŒ ๋œ๋‹ค
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(4) โญ(ํ•ต์ค‘์š”) Diagonalizable matrix A์˜ non-zero eigen value์˜ ์ˆ˜ = rank(A)

----------------sol-(4)-------------------
----------------์ฐธ๊ณ :-----------------
(diagonalizable Matrix ๋ผ๊ณ  ๋ฐ˜๋“œ์‹œ symmetric์€ ์•„๋‹ˆ๋‹ค)
-----------------------------------
A = V \gamma V^{-1} = rank(\gamma)

case 1):
\begin{bmatrix} \gamma_1 & & \\ & \gamma_2 & \\ && 0\end{bmatrix}
์ด๊ฒƒ์€ 0์ด ์žˆ์œผ๋‹ˆ ๋ง‰ํ˜€์„œ rank2

case 2):
\begin{bmatrix} \gamma_1 && \\ & \gamma_2 & \\ && \gamma_3 \\ &&& \ddots \end{bmatrix}
0์ด ๋‚˜์˜ค๊ธฐ ์ „๊นŒ์ง€ rank ๊ณ„์† ๊ฐฏ์ˆ˜ ์…ˆ
-------------------------------------------


2. feature of symmetric matrix

2-1. ํ‘œํ˜„

if A=A^T    then, symmetric matrix is diagonalizable

--------------------sol----------------------
A = V \gamma V^{-1}
A^T = V^{-T}\gamma V^{T} ์ด ๋œ๋‹ค. ๊ทธ๋Ÿผ ์ž์—ฐ์Šค๋Ÿฝ๊ฒŒ V=V^{-T}\color{red}{,} \quad\quad V^{-1}=V^T ์œผ๋กœ
๋งŒ์กฑํ•˜๋„๋ก(orthogonal matrix ์ด๋„๋ก) V๋ฅผ ์„ค์ •ํ•  ์ˆ˜ ์žˆ๋‹ค

์—ฌ๊ธฐ๊นŒ์ง€ ํ™•์ธํ–ˆ์œผ๋ฉด orthogonal Matrix๋Š” ๋ณดํ†ต Q๋กœ ํ‘œ๊ธฐํ•˜๋‹ˆ ๋‹ค์‹œ A=Q\gamma Q^T ๋กœ ๋ฐ”๊ฟ”์ ์ž
๐Ÿ’ก์ฆ‰ symmetric matrix๋Š” diagonalizableํ•˜๋ฉฐ A=Q\gamma Q^T ๊ฐ€ ๋œ๋‹ค
----------------------------------------------

2-2. ์‹ค์ƒํ™œ ์‘์šฉ

Q๊ฐ€ ๊ฐ€์ง€๋Š” ์ปฌ๋Ÿผ์„ ํ†ตํ•ด Q\gamma Q^T๋ฅผ ํ‘œํ˜„ํ•ด๋ณด์ž

(Q๋Š” 3x1,   Q^T ๋Š” 1x3์ด๋‹ค)

A=\begin{bmatrix} q_1 & q_2 & q_3 \end{bmatrix} \begin{bmatrix} \gamma_1 & & \\ & \gamma_2 & \\ && \gamma_3\end{bmatrix} \begin{bmatrix} q_1^T \\ q_2^T \\ q_3^T\end{bmatrix}

์–ด? q_1 \perp q_1^T\color{red}{,} \quad q_2 \perp q_2^T\color{red}{,} \quad q_3 \perp q_3^T๋กœ๊ตฐ

=\begin{bmatrix} \gamma_1q_1 & \gamma_1q_2 & \gamma_1q_3 \end{bmatrix} \begin{bmatrix} q_1^T \\ q_2^T \\ q_3^T \end{bmatrix} \\ \Rightarrow \gamma_1q_1q_1^T+\gamma_2q_2q_2^T+\gamma_3q_3q_3^T
์˜ค! ์ด๊ฑฐ q_1q_1^T ๋Š” ์ž์—ฐ์Šค๋Ÿฝ๊ฒŒ Rank1 Matrix๋‹ˆ๊นŒ ํ–‰๋ ฌ์€ slice๋กœ ์ชผ๊ฐ ๊ฑฐ๋„ค
Desktop View

์•„ ์ด๊ฑฐ 100ํผ ๋ฐ์ดํ„ฐ ์••์ถ•์— ์‘์šฉ๊ฐ€๋Šฅํ•˜๋‹ค

Desktop View

ex) ์‚ฌ์ง„ W๊ฐ€ 100x100 ์ด๋ผ ํ•˜๋ฉด 10000๊ฐœ์˜ ์ˆซ์ž๊ฐ€ ํ•„์š”ํ•˜๋‹ค.

์ด๊ฑธ ์ชผ๊ฐœ์„œ 5๊ฐœ๋งŒ ์“ฐ์ž \gamma_1q_1q_1^T+\gamma_2q_2q_2^T+ \cdots \gamma_{10000}q_{10000}q_{10000}^T

๊ทธ๋Ÿผ \gamma์— ๋Œ€ํ•ด 5๊ฐœ๊ฐ€ ํ•„์š”ํ•˜๊ณ  q๊ฐ€ 100x1์ด๋‹ˆ๊นŒ ์ด 5๊ฐœ ์žˆ์–ด์„œ 500
๊ทธ๋ž˜์„œ ํ•ฉํ•˜๋ฉด 505๊ฐœ๋‹ค
\color{red}{\therefore} ์ฆ‰ 10000๊ฐœ ์ค‘์—์„œ 505๊ฐœ๋ฅผ ์“ด๋‹ค๋Š” ๊ฑด๋ฐ ์„ ๋ช…ํ•˜์ง€ ์•Š๊ณ  ํ™”์งˆ์ด ๋งค์šฐ ๊ตฌ๋ฆฌ์ง€๋งŒ ์ธ์‹์€ ๋  ๊ฒƒ์ด๋‹ค


2-3. ์ƒˆ๋กœ์šด ํ•ด์„

--------------------cond(์กฐ๊ฑด)----------------------
A=A^T๋ฉด A=\gamma_1q_1q_1^T+\gamma_2q_2q_2^T+\gamma_3q_3q_3^T
(A: 3x3ํ–‰๋ ฌ์ด๊ณ ,    q_1 \perp q_2 \perp q_3)
---------------------------------------------------------

์—ฌ๊ธฐ์„œ x๋ผ๋Š” Eigen vector๊ฐ€ ์•„๋‹Œ ์ž„์˜์˜ ๋ฒกํ„ฐ๋ฅผ ํ–‰๋ ฌA์— ํ†ต๊ณผ์‹œ์ผœ decomposeํ•œ ์ƒํƒœ๋กœ ๋“ค์—ฌ๋ณด์ž

Desktop View

์–ด? q_1^Tx \quad\quad q_2^Tx \quad\quad q_3^Tx ๋Š” ๊ฐ๊ฐ x๋ž‘ ๋‚ด์ ํ•œ๊ฑฐ๋„ค??
๊ทธ๋Ÿฌ๋ฉด q_1 \quad\quad q_2 \quad\quad q_3 ๋Š” ๊ฐ๊ฐ ๋ฐฉํ–ฅ๋ฒกํ„ฐ๋‹ค
๊ทธ๋ ‡๋‹ค๋ฉด q_1q_1^Tx \quad\quad q_2q_2^Tx \quad\quad q_3q_3^Tx ๋Š” ๊ฐ๊ฐ projection์ด๋„ค ??

๐Ÿœ๏ธ๊ทธ๋ฆผ ์˜ˆ์‹œ

x๋ผ๋Š” ๋ฒกํ„ฐ๊ฐ€ ์žˆ์„ ๋•Œ ์ง๊ตํ•˜๋Š” q_1 \quad\quad q_2 \quad\quad q_3๊ฐ€ ์žˆ๋‹ค ํ•˜์ž
\color{lightgreen}{/}์„ ๋“ค์„ ์•„๋ž˜๋กœ ๋‚ด๋ฆฌ๋ฉด \color{purple}{\nearrow}๋ฒกํ„ฐ๋“ค์ด ๋Œ€์‘๋  ๊ฒƒ์ด๋‹ค
Desktop View

Desktop View



3. ๋Œ€๊ฐํ™” ํŒ๋ณ„๋ฒ•

ํŒ๋ณ„๋ฒ•์„ ๋ณด๊ธฐ์ „ ์šฐ์„  ๋Œ€๊ฐํ™” ๋ถˆ๊ฐ€๋Šฅ์‚ฌ๋ก€์™€ ๊ฐ€๋Šฅํ•œ ์‚ฌ๋ก€๋ฅผ ๋จผ์ € ๋ณด๊ณ ์ž ํ•œ๋‹ค

์˜ˆ์‹œ ใ„ฑ) ๋Œ€๊ฐํ™” ๋ถˆ๊ฐ€๋Šฅ ์‚ฌ๋ก€

A = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}   ๋Š” ๋Œ€๊ฐํ™” ๊ฐ€๋Šฅํ•œ๊ฐ€?

*์œ„์˜ ํ–‰๋ ฌ A๊ฐ€ ๋Œ€๊ฐํ™”๊ฐ€ ๊ฐ€๋Šฅํ•˜๋ ค๋ฉด ์„ ํ˜•๋…๋ฆฝ์ธ ๊ณ ์œ ๋ฒกํ„ฐ๊ฐ€ 2๊ฐœ๊ฐ€ ๋‚˜์™€์•ผํ•จ

1๏ธโƒฃ๊ณ ์œณ๊ฐ’ ๊ตฌํ•˜๊ธฐ

๊ณ ์œ ๋ฐฉ์ •์‹ \color{red}{\Rightarrow} det(\gamma I_2 - A)= det \begin{pmatrix} \gamma-2 & -1 \\ 0 & \gamma-2 \end{pmatrix} = (\gamma-2)^2 = 0 \\ \Leftrightarrow \gamma =2(์ค‘๊ทผ)

2๏ธโƒฃ๊ณ ์œ ๋ฒกํ„ฐ ๊ตฌํ•˜๊ธฐ

(2I_2 - A)v = 0 \\ \Leftrightarrow \begin{pmatrix} 0 & -1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}

์—ฌ๊ธฐ์„œ v๋Š” free variables์ธ s ํ•˜๋‚˜ ์žก๊ณ  (1,0)์„ ํ•˜๋˜ (-1,0)์„ ํ•˜๋˜ ์ƒ๊ด€์—†๋Š”๋ฐ,
(1,0)์œผ๋กœ ์„ ํƒํ•˜๊ฒ ์Šต๋‹ˆ๋‹ค.

\color{red}{\Rightarrow} ์ฆ‰ v = s\begin{pmatrix} 1 \\ 0 \end{pmatrix}   ์œผ๋กœ ๋Œ€๊ฐํ™” ๋ถˆ๊ฐ€๋Šฅ
์™œ? โ€”> ๊ณ ์œ ๊ธฐ์ € = {(1,0)}   1๊ฐœ๋ผ์„œ ์•ˆ๋œ๋‹ค!
์™œ๋ƒํ•˜๋ฉด ์„ ํ˜•๋…๋ฆฝ์ธ๊ฒŒ 2๊ฐœ๊ฐ€ ์žกํžˆ๋ ค๋ฉด ๊ณ ์œ ๊ธฐ์ €๊ฐ€ 2๊ฐœ๊ฐ€ ํ•„์š”
(ํ–‰๋ ฌ์˜ n x n)์—์„œ ํ–‰ or ์—ด๊ฐฏ์ˆ˜๋ž‘ ๋งค์น˜๋  ๊ฒƒ


์˜ˆ์‹œ ใ„ด) ๋Œ€๊ฐํ™” ๊ฐ€๋Šฅ ์‚ฌ๋ก€

์•„๊นŒ ์œ„์˜ ๋‚ด์šฉ์„ ํ† ๋Œ€๋กœ    A = \begin{pmatrix} 1 & -2 \\ 3 & -4 \end{pmatrix}   ์— ๋Œ€ํ•œ P ์ฐพ๊ธฐ

์ด๊ฑด ๋œ๋‹ค ์™œ๋ƒํ•˜๋ฉด \gamma(๊ณ ์œณ๊ฐ’) =-1   ์ผ ๋•Œ,
\rightarrow ๊ณ ์œ ๋ฒกํ„ฐ (s,s)
\rightarrow P_1\begin{pmatrix} 1 \\ 1 \end{pmatrix}   ์ผ ๋•Œ, \rightarrow ๊ณ ์œ ๋ฒกํ„ฐ (2t,3t)


\gamma(๊ณ ์œณ๊ฐ’) = -2   ์ผ ๋•Œ
\rightarrow P_2\begin{pmatrix} 2 \\ 3 \end{pmatrix}

์ฆ‰ P = P_1 P_2 = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} \\ P^{-1} = \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix} \\ P^{-1} A P = B

์ฐธ!   P_2 P_1   ๋กœ ์œ„์น˜๋ฅผ ๋ฐ”๊ฟ” ์—ด๋ฒกํ„ฐ๋“ค์„ ๋‚˜์—ดํ•˜์—ฌ ๊ณ„์‚ฐํ•ด๋„ ๋Œ€๊ฐํ™”๊ฐ€ ๋ฉ๋‹ˆ๋‹ค!




3-1 ์ค‘๋ณต๋„

๋Œ€๊ฐํ™”๊ฐ€ ๊ฐ€๋Šฅํ•œ์ง€ ํŒ๋ณ„ํ•˜๋Š” ๋˜ ๋‹ค๋ฅธ ๋ฐฉ๋ฒ•์ž…๋‹ˆ๋‹ค
๊ธฐํ•˜์  ์ค‘๋ณต๋„, ๋Œ€์ˆ˜์  ์ค‘๋ณต๋„๋ฅผ ๋น„๊ตํ•˜์—ฌ ๊ตฌํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค
์ผ๋‹จ ์š”์•ฝํ•˜๋ฉด ๊ธฐํ•˜์  ์ค‘๋ณต๋„ = ๋Œ€์ˆ˜์  ์ค‘๋ณต๋„   ์ผ ๋•Œ, ํ–‰๋ ฌ์˜ ๋Œ€๊ฐํ™”๊ฐ€ ๊ฐ€๋Šฅ ํ•ฉ๋‹ˆ๋‹ค

๊ธฐํ•˜์  ์ค‘๋ณต๋„:
๊ณ ์œ ๊ฐ’์— ๋Œ€์‘ํ•˜๋Š” ๊ณ ์œ ๊ณต๊ฐ„์˜ ์ฐจ์› ๊ฐฏ์ˆ˜

๋Œ€์ˆ˜์ ์ค‘๋ณต๋„:
๊ณ ์œ  ๋‹คํ•ญ์‹์—์„œ \gamma-\gamma_0 ๊ฐ€ ์ธ์ˆ˜๋กœ ๋‚˜ํƒ€๋‚˜๋Š” ํšŸ์ˆ˜
(๋Œ€์ˆ˜์ ์œผ๋กœ ๊ณ ์œ ๊ฐ’์ด ์ด ๋ช‡ ๊ฑฐ๋“ญ์ œ๊ณฑ์ธ์ง€?)

Desktop View

๐Ÿ‘‰๊ณ ์œ ๋ฒกํ„ฐ, ๊ณ ์œ ๊ธฐ์ €๋ฅผ ์ž˜ ๋ชจ๋ฅด๊ฒ ๋‹ค๋ฉด ์ด๊ฒƒ ํด๋ฆญ





3-2 ๋‹ฎ์Œ ๋ถˆ๋ณ€๋Ÿ‰

๋‘ ์ •์‚ฌ๊ฐํ–‰๋ ฌ A, B์— ๋Œ€ํ•˜์—ฌ B = P^{-1}AP   ๋ฅผ ๋งŒ์กฑํ•˜๋Š” ๊ฐ€์—ญํ–‰๋ ฌ P๊ฐ€ ์กด์žฌํ•˜๋ฉด
A, B๋Š” ์„œ๋กœ ๋‹ฎ์€ ํ–‰๋ ฌ์ด๋ผ ํ•˜๊ณ , ๊ธฐํ˜ธ๋กœ A~B๋ผ ํ‘œํ˜„ํ•œ๋‹ค.

์„œ๋กœ ๋‹ฎ์€ ๋‘ ํ–‰๋ ฌ์˜ ๋‹ค์Œ๊ณผ ๊ฐ™์€ ์„ฑ์งˆ๋“ค์€ ์„œ๋กœ ์ผ์น˜ํ•œ๋‹ค.
๊ทธ ์ค‘์— ์ผ๋‹จ 10๊ฐœ๋งŒ ๋ณด๊ฒ ์Šต๋‹ˆ๋‹ค

1
2
3
4
5
6
7
8
9
10
(1) ํ–‰๋ ฌ์‹
(2) ๊ฐ€์—ญ์„ฑ 
(3) rank 
(4) nullity 
(5) ๊ณ ์œ ๋‹คํ•ญ์‹(๊ณ ์œ ๋ฐฉ์ •์‹์˜ ์ขŒ๋ณ€์„ ์–˜๊ธฐํ•จ) 
(6) ๊ณ ์œณ๊ฐ’ 
(7) ๊ณ ์œ ๊ณต๊ฐ„์˜ ์ฐจ์› 
(8) ๋Œ€๊ฐ์„ฑ๋ถ„๋“ค์˜ ํ•ฉ 
(9) ๋Œ€์ˆ˜์  ์ค‘๋ณต๋„ 
(10) ๊ธฐํ•˜์  ์ค‘๋ณต๋„ 

์„œ๋กœ ๋‹ฎ์•„ ๋ณด์ด๋”๋ผ๋„, ์„ ํ˜•์‚ฌ์ƒ๋“ค์€ ์ผ๋ฐ˜์ ์œผ๋กœ ํŒŒ์•…ํ•˜๊ธฐ๊ฐ€ ๋ณต์žกํ•œ๋ฐ
์ƒ๋‹นํžˆ ๋งŽ์€ ์„ ํ˜•์‚ฌ์ƒ๋“ค์ด ์ด ํŠน์„ฑ๋“ค ์ค‘ ์ตœ์†Œ ํ•œ๊ฐœ๋ผ๋„ ๋”ฐ๋ฅผ ํ™•๋ฅ ์ด ๋†’์œผ๋‹ˆ
์„ ํ˜•์‚ฌ์ƒ๋“ค์„ ๋ถ„ํ•ดํ•˜์—ฌ ๊ฐ„์†Œํ™”๋œ ์„ ํ˜•์‚ฌ์ƒ๋“ค์—๊ฒŒ์„œ
์ € ํŠน์„ฑ๋“ค ์ค‘ ์ตœ์†Œ 1๊ฐœ ์ด์ƒ์„ ์ฐพ์„ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค
(์ฆ‰ ๋ณต์žกํ•˜๊ฒŒ ๋ง๊ณ  ์‰ฝ๊ฒŒ์‰ฝ๊ฒŒ ๋ณด์ž๋Š” ์–˜๊น๋‹ˆ๋‹ค)



4. ์ผ€์ผ๋ฆฌ-ํ•ด๋ฐ€ํ„ด ์ •๋ฆฌ

์ž„์˜์˜ ์ •์‚ฌ๊ฐํ–‰๋ ฌ A๊ณผ ๊ทธ ๊ณ ์œ ๋‹คํ•ญ์‹
f(\gamma) = det(\gamma I - A) = \sum\limits_{i=0}^n a_i\gamma^2    ์— ๋Œ€ํ•˜์—ฌ

f(A) = 0 ์ด ์„ฑ๋ฆฝํ•˜๋ฉฐ, ์ด๋ฅผ ์บ์ผ๋ฆฌ-ํ•ด๋ฐ€ํ„ด ์ •๋ฆฌ ๋ผ๊ณ  ํ•œ๋‹ค.    (๋‹จ, 0์€ ์˜ํ–‰๋ ฌ)

1
์‰ฝ๊ฒŒ ๋งํ•˜๋ฉด ๋žŒ๋‹ค์ž๋ฆฌ์— Aํ–‰๋ ฌ์„ ๋„ฃ์—ˆ๋”๋‹ˆ ์˜ํ–‰๋ ฌ์ด ๋‚˜์˜ค๋”๋ผ




์ด ๊ธ€์—์„œ๋Š” ์˜ˆ์‹œ 2๊ฐ€์ง€๋ฅผ ์ž‘์„ฑํ•ฉ๋‹ˆ๋‹ค

ex-1)

A = \begin{pmatrix} 1 & -2 \\ 3 & -4 \end{pmatrix} \\ f(\gamma) = det(\gamma I_2 - A) \\ = det \begin{pmatrix} \gamma-1 & 2 \\ -3 & \gamma+4 \end{pmatrix}

๊ทธ๋Ÿฌ๋ฉด \gamma (๊ณ ์œ ๊ฐ’)๋Š” ์•„๋ž˜์™€ ๊ฐ™์ด ๋‚˜์˜ต๋‹ˆ๋‹ค
= \gamma^2 + 3\gamma + 2


์ด๊ฑธ ์•„๋ž˜์ฒ˜๋Ÿผ ๊ณ ์น  ์ˆ˜๋„ ์žˆ์Šต๋‹ˆ๋‹ค
= a_2\gamma^2 + a_1\gamma^1 + a_0\gamma^0

์—ฌ๊ธฐ์„œ Aํ–‰๋ ฌ์„ ๋Œ€์ž…ํ•˜๋ฉด ๋‹ค์Œ๊ณผ ๊ฐ™์Šต๋‹ˆ๋‹ค
f(A) = A^2 + 3A + 2I = 0 \quad ์„ฑ๋ฆฝํ•˜๋Š”๊ฐ€?

f(A) = \begin{pmatrix} 1 & -2 \\ 3 & -4 \end{pmatrix}^2 + 3\begin{pmatrix} 1 & -2 \\ 3 & -4 \end{pmatrix}^1 + 2\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \\ \Leftrightarrow \begin{pmatrix} -5 & 6 \\ -9 & 10 \end{pmatrix} + \begin{pmatrix} 3 & -6 \\ 9 & -12 \end{pmatrix}+ \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = 0


ex-2)

ํ–‰๋ ฌ   A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} ์— ๋Œ€ํ•˜์—ฌ

A^3   ๋ฅผ ์ผ€์ผ๋ฆฌ ํ•ด๋ฐ€ํ„ด ์ •๋ฆฌ๋ฅผ ์ด์šฉํ•ด A์™€ ๋‹จ์œ„ํ–‰๋ ฌ I_2 ๋กœ์จ ํ‘œํ˜„ํ•˜์‹œ์˜ค.

ํ’€์ด:

A^3 ๋ฅผ ์ด์šฉํ•˜๋Š” ๊ฒƒ๋ณด๋‹จ,
A^2 ๋ฅผ ์ด์šฉํ•˜๋Š”๊ฒŒ ํ‘ธ๋Š”๋ฐ ๋” ์‰ฌ์šธ ์ˆ˜๋„ ์žˆ์Šต๋‹ˆ๋‹ค.

f(\gamma) = det(\gamma I_2 - A) = det \begin{pmatrix} \gamma-1 & -2 \\ -3 & \gamma-4 \end{pmatrix} = \gamma^2 -5\gamma-2 \\ \rightarrow f(A) = A^2 -5A -2I_2 =0 \quad ์„ฑ๋ฆฝํ•˜๋Š”๊ฐ€?

\rightarrow f(A) = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} - 5\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} - 2\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = 0

์ด๋ฅผ ํ†ตํ•ด A^2 = 5A+2I   ๋ผ๋Š” ๊ฒƒ์„ ํ™•์ธํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค
๊ทธ๋Ÿฌ๋ฉด ์ด์–ด์„œ ๋งˆ์ € ๊ณ„์‚ฐํ•ด๋ด…์‹œ๋‹ค

A^3 = 5A^2 + 2A \\ \Leftrightarrow A^3 = 5(5A + 2I_2) + 2I_2 \\ \Leftrightarrow A^3 = 27A + 12I_2




5. ์—ฐ์Šต๋ฌธ์ œ (3๊ฐœ) ๊ณ ์œ ๊ฐ’,๊ณ ์œ ๋ฒกํ„ฐ 2๊ฐœ, ์ผ€์ผ๋ฆฌํ•ด๋ฐ€ํ„ด 1๊ฐœ

5-1 ๐Ÿ˜€์˜ˆ์ œ1

M = \begin{pmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3\end{pmatrix}   ์˜ ๊ณ ์œณ๊ฐ’, ๊ณ ์œ ๋ฒกํ„ฐ ๊ณ ์œ ๊ธฐ์ € ๊ตฌํ•˜๊ธฐ

step 1    ๊ณ ์œณ๊ฐ’ ๊ตฌํ•˜๊ธฐ
(๊ณ ์œ ๋ฐฉ์ •์‹๋ถ€ํ„ฐ ๊ตฌํ•ฉ์‹œ๋‹ค.)

det( \gamma I_3 - M) \\ \Leftrightarrow det\begin{pmatrix} \gamma & 0 & 2 \\ -1 & \gamma - 2 & -1 \\ -1 & 0 & \gamma -3 \end{pmatrix} \\ \Leftrightarrow det = \gamma \begin{vmatrix} \gamma-2 & -1 \\ 0 & \gamma-3 \\ \end{vmatrix} - 0 \begin{vmatrix} -1 & -1 \\ -1 & \gamma-3 \\ \end{vmatrix} + 2 \begin{vmatrix} -1 & \gamma-2 \\ -1 & 0 \\ \end{vmatrix} \\ \Leftrightarrow\gamma(\gamma^2-5\gamma+6) + 2(\gamma-2) = 0 \\ \Leftrightarrow(\gamma-1)(\gamma-2)^2 = 0

์ฆ‰ ๊ณ ์œ ๊ฐ’:    \gamma = 1 or 2


step 2๊ณ ์œ ๋ฒกํ„ฐ ๊ตฌํ•˜๊ธฐ

case 1) \gamma = 1

(\gamma I_3 - M)v = 0 \\ \Leftrightarrow \begin{pmatrix} 1 & 0 & 2 \\ -1 & -1 & -1 \\ -1 & 0 & -2 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3\end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}

\Leftrightarrow \begin{pmatrix} 1 & 0 & 2 & 0\\ -1 & -1 & -1 & 0\\ -1 & 0 & -2 & 0\end{pmatrix} \color{red}{\Rightarrow} \begin{pmatrix} 1 & 0 & 2 & 0\\ 0 & -1 & 1 & 0\\ 0 & 0 & 0 & 0\end{pmatrix} \color{red}{\Rightarrow} \begin{pmatrix} 1 & 0 & 2 & 0\\ 0 & 1 & -1 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}

์—ฌ๊ธฐ์— v_1, v_2, v_3 ์„ ๊ณฑํ•˜๊ณ  v_3 ์„ S๋กœ ๋‘”๋‹ค๋ฉด Eigen value์˜ ํ•ด๋Š” ์ด๋ ‡๊ฒŒ ๋‚˜์˜ฌ๊ฒ๋‹ˆ๋‹ค.

\begin{cases} v_3 = s \\ v_2 = s \\ v_1 = -2s \end{cases} \quad \rightarrow \quad ์ฆ‰ \quad v = s\begin{pmatrix} -2 \\ 1 \\ 1\end{pmatrix}

๊ทธ๋Ÿฌ๋ฏ€๋กœ \gamma =1 \quad ์ผ ๋•Œ

๊ณ ์œ ๋ฒกํ„ฐ = (-2s, s, s) \quad s \neq0
๊ณ ์œ ๊ธฐ์ € = \{(-2,1,1)\}



case 2) \gamma = 2

(2I_3 - M)v = 0 \\ \Leftrightarrow \begin{pmatrix} 2 & 0 & 2 \\ -1 & 0 & -1 \\ -1 & 0 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 1 & 0 & 1 & |0\\ 0 & 0 & 0 & |0 \\ 0 & 0 & 0 & |0\end{pmatrix}


\begin{cases} x \quad + z = 0 \\ \quad 0y \quad\quad = 0 \\ \quad\quad 0z \quad = 0 \end{cases}
์ด๋ ‡๊ฒŒ ํ’€๋ฉด x์™€ z๋Š” ์ž์œ ๋ณ€์ˆ˜๋กœ ์„ ํƒํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค
์ฆ‰, y=1์ผ ๋•Œ์˜ ๊ณ ์œ ๋ฒกํ„ฐ๋Š” ๋‹ค์Œ๊ณผ ๊ฐ™์Šต๋‹ˆ๋‹ค
\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}
์ด์–ด์„œ ํ’€๋ฉด ์•„๋ž˜์™€ ๊ฐ™์€ ์‹์ด ๋‚˜์˜ต๋‹ˆ๋‹ค


v = t \begin{pmatrix} 0 \\ 1\\ 0 \end{pmatrix} + r \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}
์—ฌ๊ธฐ์„œ ๊ณ ์œ ๊ธฐ์ €๋Š” ์•„๋ž˜์™€ ๊ฐ™์Šต๋‹ˆ๋‹ค
\begin{pmatrix} 0 \\ 1\\ 0 \end{pmatrix},    \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}
๊ณ ์œ ๋ฒกํ„ฐ๋Š” ์•„๋ž˜์™€ ๊ฐ™์Šต๋‹ˆ๋‹ค
\begin{pmatrix} 0 \\ t\\ 0 \end{pmatrix},    \begin{pmatrix} -r \\ 0 \\ r \end{pmatrix}

1
2
3
4
v_2๊ฐ€ ์˜ํ–‰๋ ฌ์ด๋ผ ๋จผ์ € free variables๋กœ t๋ฅผ ์„ค์ •ํ•ด์ฃผ๊ณ  ๋‚˜๋จธ์ง€๋ฅผ r๋กœ ๊ตฌํ•ฉ๋‹ˆ๋‹ค
์–ด์ฐจํ”ผ ํ•ด๊ณต๊ฐ„์€ 0์ด ๋‚˜์™€์•ผ ํ•ฉ๋‹ˆ๋‹ค
v_3๊ณผ v_1๋Š” ๊ฐ™์€๋ฒกํ„ฐ๋กœ ์ค‘๋ณต์„ ์ œ๊ฑฐํ• ๊ฒธ v_3 + v+1 = 0์„ ํ†ตํ•ด ํ•ฉ์ณ์ฃผ๊ณ 
์ฆ‰ free variables 2๊ฐœ๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ v์˜ ํ•ด๊ณต๊ฐ„์€ ์œ„์™€ ๊ฐ™์ด ๋‚˜์˜ต๋‹ˆ๋‹ค

free variables์— ๋Œ€ํ•ด ์ž˜ ๋ชจ๋ฅด๊ฒ ์œผ๋ฉด ํด๋ฆญ โ€”> โœ

๊ทธ๋Ÿฌ๋ฏ€๋กœ \gamma = 2     ์ผ๋•Œ๋Š”
๊ณ ์œ ๋ฒกํ„ฐ:    (-r, t, r)
๊ณ ์œ ๊ธฐ์ €:    {(0,1,0), (-1,0,1)}

๊ฒฐ๋ก :

์ฆ‰ ์ด 3x3ํ–‰๋ ฌ M = \begin{pmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3\end{pmatrix} ์— ๋Œ€ํ•ด

\gamma(๊ณ ์œ ๊ฐ’) = 1   ์ผ๋•Œ๋Š”
๊ณ ์œ ๊ธฐ์ €๊ฐ€ ์›์†Œ 1๊ฐœ์ธ {(-2,1,1)}

\gamma(๊ณ ์œ ๊ฐ’) = 2   ์ผ๋•Œ๋Š”
๊ณ ์œ ๊ธฐ์ €๊ฐ€ ์›์†Œ 2๊ฐœ์ธ {(0,1,0), (-1,0,1)}



5-2 ์˜ˆ์ œ2

ํ–‰๋ ฌ A = \begin{pmatrix} 0 & -3 & -3 \\ 1 & 4 & 1 \\ -1 & -1 & 2\end{pmatrix}   ์— ๋Œ€ํ•ด ๋‹ค์Œ ๋ฌผ์Œ์— ๋‹ตํ•˜์‹œ์˜ค.

(1)   A๋ฅผ ๋Œ€๊ฐํ™”ํ•˜๋Š” ํ–‰๋ ฌ P๋ฅผ ๊ตฌํ•˜๊ณ ,
๋Œ€๊ฐํ–‰๋ ฌ   B = P^{-1}AP ๋ฅผ ๊ตฌํ•˜์‹œ์˜ค

(2)   ๋‘ํ–‰๋ ฌ A, B์— ๋Œ€ํ•ด ๋ณธ๋ฌธ์— ์ œ์‹œ๋œ 10๊ฐ€์ง€ ๋‹ฎ์Œ ๋ถˆ๋ณ€๋Ÿ‰์„ ๊ฐ๊ฐ ํ™•์ธํ•˜์‹œ์˜ค ๐ŸŽจ์—ฌ๊ธฐ ํด๋ฆญํ•ด์„œ ํ™•์ธ



ํ’€์ด

(1)-ใ„ฑ ๊ณ ์œณ๊ฐ’ ๊ตฌํ•˜๊ธฐ

det(\gamma I_3 -A) \\ =det \begin{pmatrix} \gamma & 3 & 3 \\ -1 & \gamma-4 & -1 \\ 1 & 1 & \gamma-2\end{pmatrix} \\ \Leftrightarrow det = \gamma \begin{vmatrix} \gamma-4 & -1 \\ 1 & \gamma-2 \\ \end{vmatrix} - 3 \begin{vmatrix} -1 & -1 \\ 1 & \gamma-2 \\ \end{vmatrix} + 3 \begin{vmatrix} -1 & \gamma-4 \\ 1 & 1 \\ \end{vmatrix} \\ \Leftrightarrow \gamma(\gamma-3)^2= 0 \\ \rightarrow \gamma = 0 \quad or \quad 3

์ฆ‰ Eigenvalue (\gamma) = 0 \quad or \quad 3


(1)-ใ„ด

\gamma =0   ์ผ ๋•Œ

\begin{pmatrix} 0 & 3 & 3 & |0 \\ -1 & -4 & -1 & |0 \\ 1 & 1 & -2 & |0 \end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 1 & 1 & -2 & 0 \\ 0 & -3 & -3 & 0 \\ 0 & 3 & 3 & 0 \end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 1 & 0 & -3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}

u๋กœ ํ–‰๋ ฌ๋ฐฉ์ •์‹์„ ๋‚˜ํƒ€๋‚ด๋ณด๊ฒ ์Šต๋‹ˆ๋‹ค.     u_1 -3u_3 = 0 \\ u_2 + u_3 = 0

u_3 ๋ฅผ ๋งค๊ฐœ๋ณ€์ˆ˜์ธ t๋กœ ํ‘œํ˜„ํ•˜๋‹ˆ ๋‚˜๋จธ์ง€
u_1, u_2 ๋„ ํ‘œํ˜„์ด ๊ฐ€๋Šฅํ•˜๋”๋ผ

๊ทธ๋Ÿฌ๋ฏ€๋กœ u = t \begin{vmatrix} 3 \\ -1 \\ 1\end{vmatrix}

์ฆ‰ \gamma =0 ์ผ ๋•Œ ๊ณ ์œ ๋ฒกํ„ฐ์ธ u์˜ ์„ฑ๋ถ„์€

\begin{vmatrix} 3t \\ -t \\ t \end{vmatrix} ๊ฐ€ ๋˜๋”๋ผ


(1)-ใ„ท

\gamma = 3   ์ผ ๋•Œ

\begin{pmatrix} 3 & 3 & 3 & |0 \\ -1 & -1 & -1 & |0 \\ 1 & 1 & 1 & |0 \end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}

์ด๋ฒˆ์—๋Š” s์™€ r์ด๋ผ๋Š” free variables๋ฅผ ์„ค์ •ํ•˜๊ฒ ์Šต๋‹ˆ๋‹ค
v = s\begin{vmatrix} ? \\ ? \\ ? \end{vmatrix} + r\begin{vmatrix} ? \\ ? \\ ? \end{vmatrix}

๋งํฌ โ€”> ์™œ free variables ์„ค์ •ํ–ˆ๋Š”์ง€ ๋ชจ๋ฅด๊ฒ ๋‹ค๋ฉด ์—ฌ๊ธฐ ํด๋ฆญ

๋ณด์•„ํ•˜๋‹ˆ ์ฒซ๋ฒˆ์งธ ์„ฑ๋ถ„(1๋ฒˆํ–‰)๊ณผ๋‹ฌ๋ฆฌ 2, 3๋ฒˆ์งธ ์„ฑ๋ถ„๋“ค์€ 0์ด๋„ค์š”.
๊ทธ์— ๋Œ€ํ•ด 2๋ฒˆ์งธ ์„ฑ๋ถ„์— ๋Œ€ํ•ด ๋ณผ ๋•Œ,   3๋ฒˆ์งธ ์„ฑ๋ถ„ = 0
3๋ฒˆ์งธ ์„ฑ๋ถ„์— ๋Œ€ํ•ด ๋ณผ ๋•Œ,   2๋ฒˆ์งธ ์„ฑ๋ถ„ = 0
์ด๋ฅผ ๋‚˜ํƒ€๋‚ด๋ฉด ์•„๋ž˜์™€ ๊ฐ™์Šต๋‹ˆ๋‹ค.

v = s\begin{vmatrix} -1 \\ 1 \\ 0 \end{vmatrix} + r\begin{vmatrix} -1 \\ 0 \\ 1 \end{vmatrix}

์—ฌ๊ธฐ์„œ ๊ณ ์œ ๊ธฐ์ €๋Š” \begin{vmatrix} -s \\ s \\ 0 \end{vmatrix} + \begin{vmatrix} -r \\ 0 \\ r \end{vmatrix}    ์ด 2๊ฐœ์˜ ์„ฑ๋ถ„์œผ๋กœ ๋งŒ๋“ค ์ˆ˜ ์žˆ๋Š” ์ง‘ํ•ฉ๋“ค์„ ์˜๋ฏธํ•ฉ๋‹ˆ๋‹ค.


(1)-ใ„น

P = \begin{pmatrix} 3 & -1 & -1 \\ -1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}   ์ผ ๋•Œ

P = A๋ฅผ ๋Œ€๊ฐํ™”ํ•˜๋Š” ํ–‰๋ ฌ
์—ฌ๊ธฐ์„  P^{-1} ๋„ ํ•„์š”ํ•œ๋ฐ ๊ฐ€์šฐ์Šค ์†Œ๊ฑฐ๋ฒ•์„ ํ†ตํ•ด ์ง„ํ–‰ํ•˜๊ฒ ์Šต๋‹ˆ๋‹ค.

๋ฐฉ๋ฒ•์€ Pํ–‰๋ ฌ ํฌ๊ธฐ๋งŒํผ ์šฐ์ธก์— ๋‹จ์œ„ํ–‰๋ ฌ์„ ์ด์–ด์ค๋‹ˆ๋‹ค
๊ทธ๋ฆฌ๊ณ  ์ขŒ์ธก ํ–‰๋ ฌ์„ ๊ธฐ์•ฝํ–‰ ์‚ฌ๋‹ค๋ฆฌ๊ผด๋กœ ๋งŒ๋“ค๋ฉด ๋ฉ๋‹ˆ๋‹ค
P^{-1} = \begin{pmatrix} 3 & -1 & -1 & |1 & 0 & 0 \\ -1 & 1 & 0 & |0 & 1 & 0\\ 1 & 0 & 1 & |0 & 0 & 1\end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 3 & -1 & -1 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 1\end{pmatrix}

์ด๋ ‡๊ฒŒ ๋ณด๋‹ˆ 3ํ–‰์˜ 1์—ด์ด ์„ ๋„์›์†Œ๋ผ์„œ ์ฒซ๋ฒˆ์งธํ–‰์œผ๋กœ ์˜ฌ๋ฆฌ๊ณ  3ํ–‰์€ ์•„๋ž˜๋กœ ๋‚ด๋ ค์˜จ ํ›„์—
๊ฐ๊ฐ ํ–‰๋“ค์„ ์—ฐ์‚ฐํ•ด ์†Œ๊ฑฐ ํ•ด์ค๋‹ˆ๋‹ค

\Leftrightarrow \begin{pmatrix} 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 \\ 0 & -1 & -4 & 1 & 0 & -3 \end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 \\ 0 & 0 & -3 & 1 & 1 & -2 \end{pmatrix} \\ P^{-1} = \begin{pmatrix} 1 & 0 & 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & 1 & 0 & \frac{1}{3} & \frac{4}{3} & \frac{1}{3} \\ 0 & 0 & 1 & -\frac{1}{3} & -\frac{1}{3} & \frac{2}{3} \end{pmatrix} \\ \Leftrightarrow P^{-1} = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 4 & 1 \\ -1 & -1 & 2\end{pmatrix}

์ด์ œ ๋ณธ์‹์— ๋Œ€์ž…ํ•ด๋ด…๋‹ˆ๋‹ค
B = P^{-1}AP

= \frac{1}{3} \begin{pmatrix} 0 & 0 & 0 \\ 3 & 12 & 3 \\ -3 & -3 & 6 \end{pmatrix} \begin{pmatrix} 3 & -1 & -1 \\ -1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} \\ = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}


(2) ๋งํฌ

์ด๊ฒƒ์„ ์ฐธ๊ณ ํ•˜์—ฌ ์ฆ๋ช…

A ํ–‰๋ ฌ,    Bํ–‰๋ ฌ(A๋ฅผ ๋Œ€๊ฐํ™”์‹œํ‚จ ํ–‰๋ ฌ)
์„ ์•„๋ž˜์™€ ๊ฐ™์ด ๊ตฌํ–ˆ์Šต๋‹ˆ๋‹ค.
A = \begin{pmatrix} 0 & -3 & -3 \\ 1 & 4 & 1 \\ -1 & -1 & 2 \end{pmatrix} \quad\quad\quad B = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}

1. ํ–‰๋ ฌ์‹
Bํ–‰๋ ฌ์€ det = 0 ๋‚˜์˜ค๋Š”๊ฒŒ ๋„ˆ๋ฌด ์ž๋ช…ํ•˜๋‹ค.
๊ทธ๋Ÿฌ๋ฉด Aํ–‰๋ ฌ๋„ ๊ณผ์—ฐ 0์ด ๋‚˜์˜ฌ๊นŒ?
(Aํ–‰๋ ฌ์„ 1์—ด ๋ฐฉํ–ฅ์œผ๋กœ ๊ณ„์‚ฐํ–ˆ์Œ)
detA = 0 \begin{vmatrix} 4 & 1 \\ -1 & 2 \\ \end{vmatrix} -1 \begin{vmatrix} -3 & -3 \\ -1 & 2 \\ \end{vmatrix} -1 \begin{vmatrix} -3 & -3 \\ 4 & 1 \\ \end{vmatrix} = -1(-6-3)-(-3+12) = 0 \\ detB = 0
์ด๋กœ์จ ๋‘˜๋‹ค ๊ฐ™์Šต๋‹ˆ๋‹ค


2. ๊ฐ€์—ญ์„ฑ
detA = 0,     detB = 0
์ฆ‰ ๋‘˜๋‹ค ์—ญํ–‰๋ ฌ์ด ์กด์žฌํ•˜์ง€ ์•Š์•„ ๋น„๊ฐ€์—ญ์„ฑ


3. rank
rankB = 2

Aํ–‰๋ ฌ์„ ๊ธฐ์•ฝํ–‰์‚ฌ๋‹ค๋ฆฌ๊ผด๋กœ ๋ณ€ํ™˜ํ•˜๋ฉด rankA๋„ ๊ตฌํ•  ์ˆ˜ ์žˆ์Œ
A = \begin{pmatrix} 0 & -3 & -3 \\ 1 & 4 & 1 \\ -1 & -1 & 2 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 4 & 1 \\ 0 & 3 & 3 \\ 0 & -3 & -3 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & -3 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix}

์ฆ‰ rankA = 2


4. Nullity
nullityA = n-rankA = nullityB
3-2 = 1


5. ๊ณ ์œ ๋‹คํ•ญ์‹
\gamma(\gamma-3)^2= 0, \quad\quad\quad B =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}
Bํ–‰๋ ฌ์˜ ๋Œ€๊ฐ์„ฑ๋ถ„๊ณผ ์ขŒ์ธก์˜ ๊ณ ์œ ๋‹คํ•ญ์‹์„ ๋ณด๋‹ˆ ์„œ๋กœ ์ผ์น˜ํ•ฉ๋‹ˆ๋‹ค


6. ๊ณ ์œณ๊ฐ’
\gamma = 0 \quad or \quad 3


7. ๊ณ ์œ ๊ธฐ์ € ์ฐจ์›
\gamma = 0   ์ผ ๋•Œ 1๊ฐœ

\gamma = 3   ์ผ ๋•Œ 2๊ฐœ


8. ๋Œ€๊ฐ์„ฑ๋ถ„ํ•ฉ
Aํ–‰๋ ฌ:
tr(A) = 0+4+2 = 6

Bํ–‰๋ ฌ:
tr(B) = 0+3+3 = 6


9. ๋Œ€์ˆ˜์  ์ค‘๋ณต๋„
\gamma ์˜ ๊ณ„์ˆ˜
\gamma=0 ์ผ ๋•Œ 1
\gamma=3 ์ผ ๋•Œ, 2

10. ๊ธฐํ•˜์  ์ค‘๋ณต๋„
๋ง ๊ทธ๋Œ€๋กœ ๊ธฐ์ €์˜ ์›์†Œ ๊ฐฏ์ˆ˜
๋งํฌ ์ฐธ์กฐ




5-3 ์˜ˆ์ œ3

ํ–‰๋ ฌ M =\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -3 & 3 \end{pmatrix}    ์— ๋Œ€ํ•˜์—ฌ ํ–‰๋ ฌ
3M^5-5M^4 ๋ฅผ ์ผ€์ผ๋ฆฌ ํ•ด๋ฐ€ํ„ด ์ •๋ฆฌ๋ฅผ ์ด์šฉํ•ด ๊ตฌํ•˜์‹œ์˜ค.

1
2
์ผ€์ผ๋ฆฌ ํ•ด๋ฐ€ํ„ด ์ •๋ฆฌ
ํ•„์š”ํ•œ๊ฒƒ: -๊ณ ์œ ๋‹คํ•ญ์‹-


step 1    ๊ณ ์œ ๋‹คํ•ญ์‹ ์ฐพ๊ธฐ
f(\gamma) = det(\gamma I_3 -M) \\ det = \begin{pmatrix} \gamma & -1 & 0 \\ 0 & \gamma & -1 \\ -1 & 3 & \gamma-3 \end{pmatrix}
1์—ด๋กœ det ์ •๋ฆฌํ•˜๋ฉด ๋  ๋“ฏ
detM = \gamma \begin{vmatrix} \gamma & -1 \\ 3 & \gamma-3 \\ \end{vmatrix} -0 \begin{vmatrix} -1 & 0 \\ 3 & \gamma-3 \\ \end{vmatrix} -1 \begin{vmatrix} -1 & 0 \\ \gamma & -1 \\ \end{vmatrix} \\ \Leftrightarrow \gamma^3-3\gamma^2+3\gamma-1 \\ \Leftrightarrow M^3-3M^2+3M-I_3 = 0

์ด๊ฒƒ์„ M^3 ์— ๋Œ€ํ•ด ์ •๋ฆฌํ•˜๋ฉด ์•„๋ž˜์™€ ๊ฐ™์ด ๋ฉ๋‹ˆ๋‹ค.

M^3 = 3M^2-3M+I_3



step 2    ์ˆ˜์‹ ๋ณ€ํ™˜ ์‘์šฉ
๋ฐฉ๊ธˆ ์œ„์˜ ์‹์— M์„ ๊ณฑํ•˜๋ฉด ์•„๋ž˜์ฒ˜๋Ÿผ ๋ฉ๋‹ˆ๋‹ค.
M^4 = 3M^3 - 3M^2 + M

์ด ์‹์—์„œ M^3 ์— ๋Œ€ํ•ด ๊ณ„์‚ฐํ•œ ๊ฒƒ์„ ๋Œ€์ž…ํ•˜๋ฉด ๋ฉ๋‹ˆ๋‹ค
๊ณ„์‚ฐํ•˜๋ฉด ์ด๋ ‡๊ฒŒ ์ •๋ฆฌ๊ฐ€ ๊ฐ€๋Šฅํ•ฉ๋‹ˆ๋‹ค
M^4 = 3(3M^2 - 3M + I_3) -3M^2 + M \\ \Leftrightarrow M^4 = 6M^2 -8M +3 I_3 \\ M^5 = MM^4 = M(6M^2 -8M +3 I_3)
\color{red}{\Rightarrow} (6M^3 -8M^2 +3M)
์ด ์‹์— ์•„๊นŒ๊ตฌํ–ˆ๋˜ M^3 ์— ๋Œ€ํ•œ ์ˆ˜์‹์„ ๋Œ€์ž…ํ•˜๋ฉด ๋ฉ๋‹ˆ๋‹ค

M^5 = 6(3M^2-3M+I_3) - 8M^2 + 3M \\ \Leftrightarrow M^5 = 10M^2 - 15M + 6I_3

์ด๋กœ์จ M^5 \quad M^4 ๋ฅผ ๋‘˜๋‹ค ๊ตฌํ–ˆ์œผ๋‹ˆ ์ด์ œ ๋ฌธ์ œ์— ๋งž์ถฐ ๊ฐ๊ฐ ๋ณ€ํ˜•์‹œํ‚ต์‹œ๋‹ค
\Leftrightarrow 3M^5-5M^4 \\3(10M^2 - 15M + 6I_3) \quad - \quad 5(6M^2 -8M +3 I_3) \\ \Leftrightarrow -5M + 3I_3

๊ฒฐ๊ณผ

์ค€์‹:    -5M + 3I_3 = \begin{pmatrix} 3 & -5 & 0 \\ 0 & 3 & -5 \\ -5 & 15 & -12 \end{pmatrix}


์ฐธ๊ณ 

  1. [์„ ํ˜•๋Œ€์ˆ˜ํ•™] 5๊ฐ•. ๊ณ ์œณ๊ฐ’๊ณผ ๋Œ€๊ฐํ™”
  2. ํ˜ํŽœํ•˜์ž„     [์„ ๋Œ€] 5-2๊ฐ•. ๊ณ ์œณ๊ฐ’ ๋ถ„ํ•ด (Eigendecomposition) ์˜ ๋ชจ๋“  ๊ฒƒ!
  3. [์žฅํ™ฉ์ˆ˜ํ•™]    ๊ณ ์œ ์น˜ ๋ฐ ๊ณ ์œ ๋ฒกํ„ฐ
  4. [์žฅํ™ฉ์ˆ˜ํ•™]    ๊ณ ์œ ์น˜ ๋ฐ ๊ณ ์œ ๋ฒกํ„ฐ์˜ ์„ฑ์งˆ
  5. [์žฅํ™ฉ์ˆ˜ํ•™]    ๋‹ฎ์€ ๋ฐ ๋Œ€๊ฐํ™” ํ–‰๋ ฌ
  6. [๊ณต๋Œ์ด์˜ ์ˆ˜ํ•™์ •๋ฆฌ๋…ธํŠธ (Angeloโ€™s Math Notes)]     ๊ณ ์œณ๊ฐ’๊ณผ ๊ณ ์œ ๋ฒกํ„ฐ
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3D GIF