"""¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$17bce861-3ef6-4050-bc79-67da0ea689b8„§cell_idÙ$17bce861-3ef6-4050-bc79-67da0ea689b8¤codeÙ'md""" (ì˜ˆì œ2) ë‘ë²ˆì§¸ ê³ ìœ ê°’ """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$6f9a1a8e-717b-4953-804a-e8647743fbd4„§cell_idÙ$6f9a1a8e-717b-4953-804a-e8647743fbd4¤codeÙ!md""" ### ì§€ë‚œì‹œê°„ ìš”ì•½ """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$71dbec22-42b4-4cbf-9e86-e8952ec127c4„§cell_idÙ$71dbec22-42b4-4cbf-9e86-e8952ec127c4¤codeÙ3md"c1 $(@bind c1 Slider(-5:0.1:5,show_value=true))"¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÂÙ$fa178a75-8119-4d2c-9f63-94420591e7a5„§cell_idÙ$fa178a75-8119-4d2c-9f63-94420591e7a5¤codeÙ`md""" (1) ìž„ì˜ì˜ ë§¤íŠ¸ë¦ìŠ¤ ${\bf X}_{n\times p}$ëŠ” í•ìƒ SVDë¶„í•´ê°€ ê°€ëŠ¥í•˜ë‹¤. """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$713775f6-ccce-4b7a-bda3-e828aebcd6a1„§cell_idÙ$713775f6-ccce-4b7a-bda3-e828aebcd6a1¤codeÚRmd""" (2) ${\bf X}$ì˜ v-martrixë¥¼ êµ¬í•˜ëŠ” ë°©ë²•ì´ ë‹¤ì–‘í•˜ë‹¤. - ë°©ë²•1: ${\bf X}$ì˜ svdë¥¼ ì´ìš©í•˜ì—¬ ì§ì ‘êµ¬í•œë‹¤. - ë°©ë²•2: ${\bf X}'{\bf X}$ì— svdë¥¼ ìˆ˜í–‰í•œë‹¤. ê·¸ëŸ¬ë©´ ì‹ ê¸°í•˜ê²Œë„ ${\bf X}'{\bf X}$ì˜ v-matrixì™€ u-matrixëŠ” í•ìƒ ì„œë¡œ ê°’ì´ ê°™ê²Œ ë‚˜ì˜¨ë‹¤. (ì™œ?) ê·¸ë¦¬ê³ ë” ì‹ ê¸°í•˜ê²Œë„ ì´ë•Œì˜ ${\bf X}'{\bf X}$ì˜ v-matrixsë¥¼ (í˜¹ì€ u-matrixë¥¼) ${\bf X}$ì˜ v-matrixë¼ê³ ì£¼ìž¥í• ìˆ˜ ìžˆë‹¤. (ì™œ?) - ë°©ë²•3: ${\bf X}'{\bf X}$ì˜ ê³ ìœ ë²¡í„°í–‰ë ¬ì„ êµ¬í•˜ë©´ ê·¸ í–‰ë ¬ì„ ${\bf X}$ì˜ v-matrixë¼ê³ ì£¼ìž¥í• ìˆ˜ ìžˆë‹¤. (ì™œ?) """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$f32f1283-1689-4372-b89b-697604efe6fc„§cell_idÙ$f32f1283-1689-4372-b89b-697604efe6fc¤codeÙümd""" (ì˜ˆì œ3) ì˜ˆì œ1,2ì˜ ê³ ìœ ë²¡í„°ë¡œ ì¡°í•©í• ìˆ˜ ìžˆëŠ” (=ê³ ìœ ë²¡í„°ë“¤ì˜ ì¡°í•©ìœ¼ë¡œ í™•ìž¥í• ìˆ˜ ìžˆëŠ”, ê³ ìœ ë²¡í„°ë¡œ spaní•˜ëŠ”) ì ë“¤ì„ ì‹œê°í™” í•˜ë¼. ì˜ˆì œ1ì˜ ë§¤íŠ¸ë¦ìŠ¤ë¥¼ ìž„ì˜ë¡œ ë°”ê¿”ë³´ë©´ì„œ ê´€ì°°í•´ë³´ë¼. """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$006740e0-f12c-4312-b181-30d5e6f82ec3„§cell_idÙ$006740e0-f12c-4312-b181-30d5e6f82ec3¤codeÙ%md""" ## Eigenvalue Decomposition """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$c1e1aded-6005-4b01-bab1-8f75a3811dc4„§cell_idÙ$c1e1aded-6005-4b01-bab1-8f75a3811dc4¤codeÙÉmd""" (2) ${\bf X}_{n \times p}$ê°€ ì´ë¯¸ì§€ì¼ë•Œ: SVDë¥¼ ì´ìš©í•˜ë©´ ${\bf X}_{n\times p} \approx {\bf \hat{X}}_{n\times p}$ ì¸ ${\bf \hat{X}}_{n\times p}$ë¥¼ êµ¬í• ìˆ˜ ìžˆë‹¤. (ì••ì¶•í•˜ë‹ˆ) """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$80393f7b-c7d2-4972-9ed6-6d06439760ee„§cell_idÙ$80393f7b-c7d2-4972-9ed6-6d06439760ee¤codeÙ9md""" (ì˜ˆì œ1) 2ê°œì˜ ê³ ìœ ê°’ì´ ê²¹ì¹˜ëŠ” ì˜ˆì œ """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$fe5876ba-30f9-470d-bd42-4c188fa580f3„§cell_idÙ$fe5876ba-30f9-470d-bd42-4c188fa580f3¤codeÚïmd""" `-` ìž„ì˜ì˜ ì •ì‚¬ê°í–‰ë ¬ ${\bf A}_{n\times n}$ì— ëŒ€í•˜ì—¬ ì–´ë– í•œ ë²¡í„° ${\psi}_{n\times 1}\neq 0$ ê°€ ì ë‹¹í•œí•œ ê°’ $\lambda$ì— ëŒ€í•˜ì—¬ $${\bf A}{\psi} = \lambda \psi$$ ë¥¼ ë§Œì¡±í•˜ë©´ $\psi$ë¥¼ $\lambda$ì˜ ê³ ìœ ë²¡í„°ë¼ê³ í•˜ê³ $\lambda$ëŠ” $\psi$ì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ê°’ì´ë¼ê³ í•œë‹¤. - note: 0-ë²¡í„°ëŠ” ê³ ìœ ë²¡í„°ë¡œ ì¸ì •í•˜ì§€ ì•ŠìŒ $\to$ ê³ ìœ ê°’ì„ ì°¾ëŠ”ë°©ë²•? $\det({\bf A}-\lambda {\bf I})=0$ì„ ë§Œì¡±í•˜ëŠ” $\lambda$ë¥¼ í’€ë©´ëœë‹¤. """¨metadataƒ©show_logsÃ¨disabledÂ®skip_as_scriptÂ«code_foldedÃÙ$3a85c442-10cd-46ac-943f-6912018b008c„§cell_idÙ$3a85c442-10cd-46ac-943f-6912018b008c¤codeÙ

ì§€ë‚œì‹œê°„ê¹Œì§€ì˜ ë‚´ìš© ìš”ì•½

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(3) ${\bf X}_{n \times p}$ê°€ ë°ì´í„°í”„ë ˆìž„ì¼ë•Œ: SVDë¥¼ ì´ìš©í•˜ë©´ ${\bf X}_{n \times p}$ë³´ë‹¤ rankê°€ ìž‘ì€ ë§¤íŠ¸ë¦ìŠ¤ ${\bf Z}_{n \times q <p}$ ë¥¼ ì°¾ì„ ìˆ˜ ìžˆë‹¤. ${\bf Z}_{n\times q}$ëŠ” ì ë‹¹í•œ ë³€í™˜ ${\bf B}_{q\times p }$ì— ì˜í•˜ì—¬ ${\bf Z}_{n\times q}{\bf B}_{q\times p}={\bf \hat{X}}_{n\times p} \approx {\bf X}_{n\times p}$ ì´ ëœë‹¤ëŠ” íŠ¹ì§•ì´ ìžˆë‹¤.

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- ì˜ˆì œ1ì´ ê°€ì§ˆ ìˆ˜ ìžˆëŠ” ê³ ìœ ë²¡í„° ì¡°í•©ë“¤ì´ ì˜ˆì œ2ë³´ë‹¤ ë” í’ë¶€í•˜ë‹¤. ì´ ëŠë‚Œì„ ì¢€ ë” ìˆ˜í•™ì ìœ¼ë¡œ í‘œí˜„í• ìˆ˜ ìžˆì„ê¹Œ? $\to$ ì˜ˆì œ1ì˜ ê³ ìœ ë²¡í„°ë“¤ë¡œ ì¡°í•©í• ìˆ˜ ìžˆëŠ” (=ê³ ìœ ë²¡í„°ë“¤ì˜ ì¡°í•©ìœ¼ë¡œ í™•ìž¥í• ìˆ˜ ìžˆëŠ”, ê³ ìœ ë²¡í„°ë¡œ spaní•˜ëŠ”) ì ë“¤ì€ 2ì°¨ì› í‰ë©´ì„ ë§Œë“¤ì§€ë§Œ ì˜ˆì œ2ì˜ ê³ ìœ ë²¡í„°ë¡œ ì¡°í•©í• ìˆ˜ ìžˆëŠ” (=ê³ ìœ ë²¡í„°ë“¤ì˜ ì¡°í•©ìœ¼ë¡œ í™•ìž¥í• ìˆ˜ ìžˆëŠ”, ê³ ìœ ë²¡í„°ë¡œ spaní•˜ëŠ”) ì ë“¤ì€ 1ì°¨ì› ì§ì„ ì„ ë§Œë“ ë‹¤.

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$\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ ì´ ê³ ìœ ë²¡í„° ì´ë¯€ë¡œ $\begin{bmatrix} 0.5 \\ 0 \end{bmatrix}$, $\begin{bmatrix} -3.14 \\ 0 \end{bmatrix}$ ë“±ë„ ê³ ìœ ë²¡í„°ì´ë‹¤.

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ìˆ™ì œ

ì •ì‚¬ê°í–‰ë ¬ ${\bf A}_{2\times 2}$ê°€ 0í–‰ë ¬ì¸ ê²½ìš°ì— ${\bf A}$ì˜ ê³ ìœ ë²¡í„°ë“¤ì´ spaní•˜ëŠ” ê³µê°„ì´ ëª‡ì°¨ì›ì¸ê°€?

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ê³ ìœ ê°’ê³¼ ê³ ìœ ë²¡í„°ì˜ ì •ì˜

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ì§€ë‚œì‹œê°„ summary

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(ì˜ˆì œ1) ì²«ë²ˆì§¸ ê³ ìœ ê°’

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$\psi = \begin{bmatrix} -2 \\ 2 \end{bmatrix}$ ì—ì‹œ $\lambda=-1$ì— ëŒ€ì‘í•˜ëŠ” ${\bf A}$ì˜ ê³ ìœ ë²¡í„°ì´ë‹¤.

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$\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ ì´ ê³ ìœ ë²¡í„° ì´ë¯€ë¡œ $\begin{bmatrix} -1 \\ 0 \end{bmatrix}$, $\begin{bmatrix} -3.14 \\ 0 \end{bmatrix}$ ë“±ë„ ê³ ìœ ë²¡í„°ì´ë‹¤.

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usings

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(ì‘ìš©ë²„ì „) ì ë‹¹í•œ ë§¤íŠ¸ë¦ìŠ¤ ${\bf V}=[V_1~ V_2~ \dots ~ V_n]$ì— ëŒ€í•˜ì—¬ ${\bf c} \neq 0$ì¸ ë²¡í„° ${\bf c}= (c_1,\dots,c_n)^\top$ì´ ì¡´ìž¬í•˜ì—¬ ${\bf V}{\bf c}=0$ ë§Œì¡±í•˜ë©´ ${\bf V}$ì˜ columnë“¤ì€ ì„ í˜•ë…ë¦½ì´ ì•„ë‹ˆë¼ê³ ë³¼ ìˆ˜ ìžˆë‹¤.

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c2 -5.0

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ëª‡ ê°€ì§€ ìš©ì–´ì˜ ì•½ì† (ì±…ì— ì—†ëŠ” ìš©ì–´ë„ ìžˆìŒ)

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(ì˜ˆì œ1)

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ê³ ìœ ë²¡í„°ì˜ ì°¨ì›

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(2) ì •ì‚¬ê°í–‰ë ¬ ${\bf A}$ê°€ ê³ ìœ ê°’ë¶„í•´ê°€ ê°€ëŠ¥í• ë•Œ ${\bf A}$ì˜ ì„œë¡œë…ë¦½ì¸ ê³ ìœ ë²¡í„°ë¥¼ column-wiseí•˜ê²Œ í•©ì¹œ í–‰ë ¬ì„ ${\bf A}$ì˜ ê³ ìœ ë²¡í„°í–‰ë ¬(eigenvector matrix)ì´ë¼ê³ í•˜ìž. ê·¸ë¦¬ê³ ê³ ìœ ë²¡í„°í–‰ë ¬ì˜ each columnsì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ê°’ì„ ìˆœì„œëŒ€ë¡œ ë‚˜ì—´í•˜ì—¬ ëŒ€ê°ì„ ì˜ ì›ì†Œì— ë„£ì€ ëŒ€ê°í–‰ë ¬ì„ ê³ ìœ ê°’í–‰ë ¬(eigenvalue matrix)ë¼ê³ í•˜ìž.

ì°¸ê³ ë¡œ ${\bf X}$ì˜ ê³ ìœ ë²¡í„°í–‰ë ¬ì€ ìœ ì¼í•˜ì§€ ì•ŠìŒ. ì˜ˆë¥¼ë“¤ì–´ ë§¤íŠ¸ë¦ìŠ¤ ${\bf \Psi}$ê°€ ${\bf X}$ì˜ ê³ ìœ ë²¡í„°í–‰ë ¬ì´ë©´ $-{\bf \Psi}$ ì—ì‹œ ${\bf X}$ì˜ ê³ ìœ ë²¡í„°í–‰ë ¬ìž„.
ê³ ìœ ë²¡í„°ì˜ ìˆœì„œë¥¼ ë°”ê¿€ìˆ˜ ìžˆê¸°ì— ê³ ìœ ê°’í–‰ë ¬ë„ ìœ ì¼í•˜ì§€ëŠ” ì•ŠìŒ. í•˜ì§€ë§Œ ê³ ìœ ê°’ì„ ìˆœì„œëŒ€ë¡œ ì •ë¦¬í•œë‹¤ë©´ ìœ ì¼í•¨.

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$\lambda = -1$ì¼ë•Œ $\det ({\bf A}-\lambda {\bf I})=0$ ì´ë¯€ë¡œ $\lambda =-1$ì€ ${\bf A}$ì˜ ê³ ìœ ê°’ì´ë‹¤.

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$\lambda = 3$ì¼ë•Œ $\det ({\bf A}-\lambda {\bf I})=0$ ì´ë¯€ë¡œ $\lambda =3$ì€ ${\bf A}$ì˜ ê³ ìœ ê°’ì´ë‹¤.

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5ì›”24ì¼

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(ì˜ˆì œ3) ì²«ë²ˆì§¸ ê³ ìœ ê°’ì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ë²¡í„°

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(ì˜ˆì œ5) ë‘ë²ˆì§¸ ê³ ìœ ê°’ì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ë²¡í„°

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(ì½”ë“œì˜ˆì‹œ)

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(ì™œ?) íŠ¹ì„±ë°©ì •ì‹ì„ ë§Œì¡±í•˜ëŠ” í•˜ë‚˜ì˜ ê·¼ $\lambda^*$ë¥¼ fixí•˜ìž. ê³ ì •ëœ $\lambda^*$ì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ë²¡í„°ê°€ ì—†ë‹¤ëŠ” ì˜ë¯¸ëŠ”

$$({\bf A}-\lambda^*{\bf I})\psi =0$$

ë¥¼ ë§Œì¡±í•˜ëŠ” $\psi$ëŠ” ì˜¤ì§ $\psi=0$ ë¿ì´ë¼ëŠ” ê²ƒì„ ì˜ë¯¸í•œë‹¤. ê·¸ëŸ°ë° ì´ëŠ” ì‚¬ì‹¤ì´ ì•„ë‹ˆë‹¤. ì™œëƒí•˜ë©´ $\lambda^*$ëŠ”

$$\det({\bf A}-\lambda^*{\bf I})=0$$

ë¥¼ ë§Œì¡±í•˜ê³ ë”°ë¼ì„œ í–‰ë ¬ ${\bf A}-\lambda^*{\bf I}$ëŠ” ì—í–‰ë ¬ì´ ì—†ëŠ” í–‰ë ¬ì´ ë˜ê³ ì´ë ‡ê²Œ ë˜ë©´ ${\bf A}-\lambda^*{\bf I}$ì˜ column ë“¤ì€ ì„ í˜•ë…ë¦½ì´ ì•„ë‹ˆê²Œ ëœë‹¤. ë”°ë¼ì„œ $({\bf A}-\lambda^*{\bf I})\psi=0$ì„ ë§Œì¡±í•˜ëŠ” $\psi\neq0$ê°€ ìžˆë‹¤.

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(1) ${\bf X}={\bf U}{\bf D}{\bf V}^\top$ ì¼ë•Œ ${\bf Z}={\bf \tilde U}{\bf \tilde D}={\bf X}{\bf \tilde V}$ ì™€ ê°™ì´ êµ¬í• ìˆ˜ ìžˆë‹¤.

ì´ë•Œ ${\bf \tilde U}$, ${\bf \tilde D}$, ${\bf \tilde V}$ì˜ ì •ì˜ëŠ” ì§€ë‚œê°•ì˜ë…¸íŠ¸ ì°¸ê³

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- ì•„ëž˜ì˜ ì˜ˆì œë“¤ì„ ê´€ì°°í•˜ìž.

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ê³ ìœ ê°’ì˜ ì¡´ìž¬

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íŠ¹ì„±ë°©ì •ì‹ì˜ ê·¼ì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ë²¡í„°ì˜ ì¡´ìž¬

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(ì˜ˆì œ2)

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- ê·¸ëŸ°ë° $n$ì°¨ ë‹¤í•ì‹ì˜ í•´ê°€ ì¤‘ë³µê·¼ì¼ ìˆ˜ë„ ìžˆìœ¼ë¯€ë¡œ ${\bf A}$ê°€ ì„œë¡œ ë‹¤ë¥¸ $n$ê°œì˜ ê³ ìœ ê°’ì„ ê°€ì§ˆ í•„ìš”ëŠ” ì—†ë‹¤.

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(ë³´ì¶©í•™ìŠµ) ë²¡í„° $V_1,V_2,\dots,V_n$ì´ ì„ í˜•ë…ë¦½ì´ ì•„ë‹ˆë¼ëŠ” ì˜ë¯¸ëŠ” ì ë‹¹í•œ $c_1,c_2,\dots,c_n$ì´ ì¡´ìž¬í•˜ì—¬

$$c_1 V_1 + c_2V_2 + \dots + c_nV_n=0$$

ì„ ë§Œì¡±í•œë‹¤ëŠ” ì˜ë¯¸ì´ë‹¤. (ë‹¨, ì´ë•Œ $c_1,\dots c_n$ì´ ëª¨ë‘ 0ì€ ì•„ë‹ˆë¼ê³ ê°€ì •í•œë‹¤.)

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- "í•˜ë‚˜ì˜ ê³ ìœ ê°’ì— ë°˜ë“œì‹œ í•˜ë‚˜ì˜ ê³ ìœ ë²¡í„°ëŠ” ì¡´ìž¬í•œë‹¤"ë¼ëŠ” ë§ì€ ì‚¬ì‹¤ ë¬´í•œê°œì˜ ê³ ìœ ë²¡í„°ê°€ ì¡´ìž¬í•œë‹¤ëŠ” ê²ƒì„ ì˜ë¯¸í•œë‹¤. ì™œëƒí•˜ë©´ $\psi$ê°€ ê³ ìœ ë²¡í„°ì´ë©´ $\sqrt{2}\psi$ë„ ê³ ìœ ë²¡í„°ì´ê¸° ë•Œë¬¸ì´ë‹¤.

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(ì½”ë“œì˜ˆì‹œ)

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- ê³ ìœ ê°’ì´ ì—†ëŠ” ì •ì‚¬ê°í–‰ë ¬ì€ ì—†ë‹¤.

(ì™œ?) í–‰ë¥¼ ${\bf A}_{n\times n}$ì˜ ê³ ìœ ê°’ì´ ì—†ë‹¤ëŠ” ì˜ë¯¸ëŠ” $\det({\bf A}-\lambda {\bf I})=0$ë¥¼ ë§Œì¡±í•˜ëŠ” $\lambda$ê°€ ì—†ë‹¤ëŠ” ì˜ë¯¸ì´ë‹¤. ê·¸ëŸ°ë° ìž„ì˜ì˜ $n$ì°¨ ë‹¤í•ì‹ì˜ í•´ëŠ” í•ìƒ ì¡´ìž¬í•œë‹¤ (ëŒ€ìˆ˜í•™ì˜ ê¸°ë³¸ì •ë¦¬).

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$\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ ì´ ê³ ìœ ë²¡í„° ì´ë¯€ë¡œ $\begin{bmatrix} 0 \\ 5.3 \end{bmatrix}$, $\begin{bmatrix} 0 \\ -\sqrt{3} \end{bmatrix}$ ë“±ë„ ê³ ìœ ë²¡í„°ì´ë‹¤.

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- ì˜ˆì œ1,2 ëª¨ë‘ ë¬´í•œê°œì˜ ê³ ìœ ë²¡í„°ë¥¼ ê°€ì§€ëŠ” ê²ƒì€ ë§žì§€ë§Œ ì°¨ì›ì´ ë‹¤ë¥´ë‹¤. ì˜ˆì œ2ì˜ ê²½ìš° $\begin{bmatrix} 1 \\ 0\end{bmatrix}$ì„ basisë¡œ í•œ ë²¡í„°ë“¤ì˜ ì¡°í•©ì„ ë§Œë“¤ ìˆ˜ ìžˆì§€ë§Œ ì˜ˆì œ1ì˜ ê²½ìš° $\begin{bmatrix} 1 \\ 0\end{bmatrix}$ì„ basisë¡œ í•œ ë²¡í„°ë“¤ì˜ ì¡°í•©ì„ ë§Œë“¤ ìˆ˜ ìžˆê³ ì¶”ê°€ì ìœ¼ë¡œ $\begin{bmatrix} 0 \\ 1\end{bmatrix}$ë¥¼ basisë¡œ í•œ ë²¡í„°ë“¤ì˜ ì¡°í•©ë„ ë§Œë“¤ ìˆ˜ ìžˆê¸° ë•Œë¬¸ì´ë‹¤.

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$\psi = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ ì€ $\lambda=3$ì— ëŒ€ì‘í•˜ëŠ” ${\bf A}$ì˜ ê³ ìœ ë²¡í„°ì´ë‹¤.

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- íŠ¹ì„±ë°©ì •ì‹ì„ ë§Œì¡±í•˜ëŠ” ê·¼ $\lambda$ì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ë²¡í„°ê°€ ë°˜ë“œì‹œ í•˜ë‚˜ëŠ” ì¡´ìž¬í•œë‹¤.

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(ì˜ˆì œ2) 2ê°œì˜ ê³ ìœ ê°’ì´ ê²¹ì¹˜ëŠ” ë‘ë²ˆì§¸ ì˜ˆì œ

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(1) ${\bf X}$ì˜ svd ë¶„í•´ê¼´ë¡œ ì–»ì–´ì§€ëŠ” ê²°ê³¼ë¥¼ ê°ê° ${\bf X}$ì˜ u-matrix, d-matrix, v-matrix ë¼ê³ ë¶€ë¥´ìž.

ì°¸ê³ ë¡œ ${\bf X}$ì˜ u-matrix, v-matrix ìœ ì¼í•˜ì§€ ì•ŠìŒ.
u-matrixì™€ v-matrixì˜ columnë“¤ì˜ ìˆœì„œë¥¼ ìž„ì˜ë¡œ ë°”ê¿€ìˆ˜ ìžˆìœ¼ë¯€ë¡œ ${\bf X}$ì˜ d-matrix ì—ì‹œ ìœ ì¼í•˜ì§„ ì•ŠìŒ. ê·¸ë ‡ì§€ë§Œ ìˆœì„œë¥¼ ì •ë ¬í•œë‹¤ë©´ ìœ ì¼í•¨.

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(ì˜ˆì œ2) ë‘ë²ˆì§¸ ê³ ìœ ê°’

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ì§€ë‚œì‹œê°„ ìš”ì•½

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c1 -5.0

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(1) ìž„ì˜ì˜ ë§¤íŠ¸ë¦ìŠ¤ ${\bf X}_{n\times p}$ëŠ” í•ìƒ SVDë¶„í•´ê°€ ê°€ëŠ¥í•˜ë‹¤.

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ë°©ë²•2: ${\bf X}'{\bf X}$ì— svdë¥¼ ìˆ˜í–‰í•œë‹¤. ê·¸ëŸ¬ë©´ ì‹ ê¸°í•˜ê²Œë„ ${\bf X}'{\bf X}$ì˜ v-matrixì™€ u-matrixëŠ” í•ìƒ ì„œë¡œ ê°’ì´ ê°™ê²Œ ë‚˜ì˜¨ë‹¤. (ì™œ?) ê·¸ë¦¬ê³ ë” ì‹ ê¸°í•˜ê²Œë„ ì´ë•Œì˜ ${\bf X}'{\bf X}$ì˜ v-matrixsë¥¼ (í˜¹ì€ u-matrixë¥¼) ${\bf X}$ì˜ v-matrixë¼ê³ ì£¼ìž¥í• ìˆ˜ ìžˆë‹¤. (ì™œ?)
ë°©ë²•3: ${\bf X}'{\bf X}$ì˜ ê³ ìœ ë²¡í„°í–‰ë ¬ì„ êµ¬í•˜ë©´ ê·¸ í–‰ë ¬ì„ ${\bf X}$ì˜ v-matrixë¼ê³ ì£¼ìž¥í• ìˆ˜ ìžˆë‹¤. (ì™œ?)

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(ì˜ˆì œ3) ì˜ˆì œ1,2ì˜ ê³ ìœ ë²¡í„°ë¡œ ì¡°í•©í• ìˆ˜ ìžˆëŠ” (=ê³ ìœ ë²¡í„°ë“¤ì˜ ì¡°í•©ìœ¼ë¡œ í™•ìž¥í• ìˆ˜ ìžˆëŠ”, ê³ ìœ ë²¡í„°ë¡œ spaní•˜ëŠ”) ì ë“¤ì„ ì‹œê°í™” í•˜ë¼. ì˜ˆì œ1ì˜ ë§¤íŠ¸ë¦ìŠ¤ë¥¼ ìž„ì˜ë¡œ ë°”ê¿”ë³´ë©´ì„œ ê´€ì°°í•´ë³´ë¼.

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Eigenvalue Decomposition

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(2) ${\bf X}_{n \times p}$ê°€ ì´ë¯¸ì§€ì¼ë•Œ: SVDë¥¼ ì´ìš©í•˜ë©´ ${\bf X}_{n\times p} \approx {\bf \hat{X}}_{n\times p}$ ì¸ ${\bf \hat{X}}_{n\times p}$ë¥¼ êµ¬í• ìˆ˜ ìžˆë‹¤. (ì••ì¶•í•˜ë‹ˆ)

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(ì˜ˆì œ1) 2ê°œì˜ ê³ ìœ ê°’ì´ ê²¹ì¹˜ëŠ” ì˜ˆì œ

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- ìž„ì˜ì˜ ì •ì‚¬ê°í–‰ë ¬ ${\bf A}_{n\times n}$ì— ëŒ€í•˜ì—¬ ì–´ë– í•œ ë²¡í„° ${\psi}_{n\times 1}\neq 0$ ê°€ ì ë‹¹í•œí•œ ê°’ $\lambda$ì— ëŒ€í•˜ì—¬

$${\bf A}{\psi} = \lambda \psi$$

note: 0-ë²¡í„°ëŠ” ê³ ìœ ë²¡í„°ë¡œ ì¸ì •í•˜ì§€ ì•ŠìŒ $\to$ ê³ ìœ ê°’ì„ ì°¾ëŠ”ë°©ë²•? $\det({\bf A}-\lambda {\bf I})=0$ì„ ë§Œì¡±í•˜ëŠ” $\lambda$ë¥¼ í’€ë©´ëœë‹¤.

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${\bf Z}$ì™€ ${\bf B}$ì— ëŒ€í•œ ì•”ê¸°ì‚¬í•

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(ì˜ˆì œ4) ì²«ë²ˆì§¸ ê³ ìœ ê°’ì— ëŒ€ì‘í•˜ëŠ” ë˜ ë‹¤ë¥¸ ê³ ìœ ë²¡í„°

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- ì˜ˆì œ3,4ì˜ ê´€ì°°: $\psi$ê°€ ${\bf A}$ì˜ ê³ ìœ ë²¡í„°ì´ë¼ë©´ $-\psi, \frac{1}{\sqrt{2}}\psi,\dots$ ëª¨ë‘ ${\bf A}$ì˜ ê³ ìœ ë²¡í„°ì´ë‹¤. ê·¸ë¦¬ê³ ì´ë•Œ $\psi, -\psi, \frac{1}{\sqrt{2}}\psi$ì— ëŒ€ì‘í•˜ëŠ” ê³ ìœ ê°’ì€ ëª¨ë‘ ê°™ë‹¤.

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$\psi = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$ì€ $\lambda=-1$ì— ëŒ€ì‘í•˜ëŠ” ${\bf A}$ì˜ ê³ ìœ ë²¡í„°ì´ë‹¤.

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ì§€ë‚œì‹œê°„ê¹Œì§€ì˜ ë‚´ìš© ìš”ì•½

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