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- ì´ë¡ : 모든 ì›ì†Œê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ (1) 대ê°í™”가능하고 (2) ì‹¤ìˆ˜ì˜ ê³ ìœ ê°’ì„ ê°€ì§€ë©° (3) ê³ ìœ ë²¡í„°í–‰ë ¬ì´ ì§êµí–‰ë ¬ì´ë‹¤. 즉 모든 ì›ì†Œê°€ 실대칭행렬 ${\bf A}$는 아래와 ê°™ì´ í‘œí˜„í•  수 있다.

$${\bf A} = {\bf \Psi}{\bf \Lambda}{\bf \Psi}^\top$$

단 ${\bf \Lambda}$ì˜ ëª¨ë“  대ê°ì„  ì›ì†ŒëŠ” 실수ì´ë‹¤.

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Positive definite matrix ì •ì˜

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- ì •ì˜: ì–´ë– í•œ 매트릭스 ${\bf A}_{n\times n}$ê°€ 모든 non-zero vector ${\bf y}_{n \times 1}$ì— ëŒ€í•˜ì—¬

$${\bf y}^\top {\bf A}{\bf y} > 0$$

ì„ ë§Œì¡±í•˜ë©´ ${\bf A}$를 positive definite matrixë¼ê³  부른다.

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(예제1) 예를들면 ${\bf A}=\begin{bmatrix} 2 & 0 \\ 2 & 2 \end{bmatrix}$는 positive definite ì˜ ì •ì˜ë¥¼ 만족한다.

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(예제2) ${\bf A}=\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$ 는 positive definiteì˜ ì •ì˜ë¥¼ 만족한다.

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(예제3) 대칭행렬ì´ì§€ë§Œ 모든 ì›ì†Œê°€ 실수가 ì•„ë‹Œ 경우ì—는 대ê°í™” 불가능 í•  ìˆ˜ë„ ìžˆë‹¤.

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ì´ì „ì¦ëª…ì„ ì‚­ì œí•˜ê³  ì´ ì¦ëª…으로 수정하였습니다. ì´ ì¦ëª…으로 공부하세요!!

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  • ì´ë•Œ $\lambda=i$ ì´ê³  $\psi_1=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -i \end{bmatrix}$, $\psi_2 = \frac{1}{\sqrt{2}} \begin{bmatrix} i \\ 1 \end{bmatrix}$ ì´ë‹¤.

  • $\psi_1 i = \psi_2$ìž„ì„ ê´€ì°°í•˜ë¼. (ë”°ë¼ì„œ ê³ ìœ ë²¡í„°í–‰ë ¬ì´ full rankê°€ 아니다)

  • 즉 ì´ í–‰ë ¬ì€ ëŒ€ê°í™”ê°€ 불가능한 행렬ì´ë‹¤.

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usings

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Spectral theorem

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(왜?)

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(예제1) 0í–‰ë ¬ì€ ëª¨ë“  ì›ì†Œê°€ 실수ì´ë©° 대칭행렬ì´ë‹¤. ë”°ë¼ì„œ ì§êµëŒ€ê°í™” 가능하다.

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Positive definite and symmetric matrix

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5ì›”31ì¼

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- 요약: ${\bf A}$ê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ì´ë©´ 아래가 ë™ì¹˜ì´ë‹¤.

  • 행렬 ${\bf A}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì´ë‹¤.

  • 행렬 ${\bf A}$ê°€ positive definite matrixì´ë‹¤.

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  • note: 참고로 ì´ë•Œ ${\bf A}$ì˜ ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì¸ ê²ƒì€ ì•„ë‹ˆë‹¤.

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- 요약: ${\bf A}$ê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ì´ë©´ 아래가 ë™ì¹˜ì´ë‹¤.

  • 행렬 ${\bf A}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì´ 0ë˜ëŠ” 양수ì´ë‹¤.

  • 행렬 ${\bf A}$ê°€ positive semidefinite matrixì´ë‹¤.

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Positive definite matrix

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숙제

ê³ ìœ ë²¡í„°í–‰ë ¬ì´ ì§êµí–‰ë ¬ì´ê³  모든 ê³ ìœ ê°’ì´ ì‹¤ìˆ˜ê°€ ë˜ëŠ” 매트릭스 ${\bf A}_{4\times 4}$ 를 구해보ë¼.

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(why?)

$${\bf y}'{\bf A}{\bf y}=(y_1+y_2)^2+y_1^2+y_2^2>0$$

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- 관찰1: ì–´ë– í•œ 실대칭행렬 ${\bf A}_{n\times n}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì´ë©´ ${\bf A}$는 positive definite matrixê°€ ëœë‹¤. (즉 모든 non-zero vector ${\bf y}$ì— ëŒ€í•´ì„œ ${\bf y}^\top {\bf A}{\bf y}>0$ì„ ë§Œì¡±í•œë‹¤.)

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모든 ìž„ì˜ì˜ ${\bf y}$ì— ëŒ€í•˜ì—¬ 아래가 성리하므로.

$${\bf y}^\top{\bf A}{\bf y}={\bf y}^\top{\bf \Psi}{\bf \Lambda}{\bf \Psi}^\top{\bf y}={\bf x}^\top {\bf \Lambda}{\bf x}=\sum_{i=1}^{n}x_i^2\lambda_i>0$$

단 여기ì—ì„œ ${\bf x}={\bf \Psi}^\top{\bf y}$ ì´ë‹¤.

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(예제2) 아무렇게나 ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ì„ ë§Œë“¤ê¸°ë§Œí•˜ë©´ ì§êµëŒ€ê°í™”ê°€ 가능하다.

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- 관찰2: ì–´ë– í•œ 실대칭행렬 ${\bf A}$ê°€ positive definite matrix ì´ë©´ (즉 모든 non-zero vector ${\bf y}$ì— ëŒ€í•˜ì—¬ ${\bf y}^\top {\bf A}{\bf y}>0$ì´ ì„±ë¦½í•œë‹¤ë©´) ${\bf A}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì€ ì–‘ìˆ˜ì´ë‹¤.

¤mime©text/html¬rootassigneeÀ²last_run_timestampËAÙé"{aß°persist_js_state·has_pluto_hook_features§cell_idÙ$706c2e23-626e-44e0-a443-70023ffbccc6¹depends_on_disabled_cells§runtimeÎ%Bµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$1bcbd243-7088-4133-a8a9-6dab8d37001cŠ¦queued¤logs§running¦output†¤bodyÙ›

  • note: 참고로 ì´ë•Œ ${\bf A}$는 ëŒ€ì¹­í–‰ë ¬ì´ ì•„ë‹ˆë‹¤.

¤mime©text/html¬rootassigneeÀ²last_run_timestampËAÙé"{a6Ê°persist_js_state·has_pluto_hook_features§cell_idÙ$1bcbd243-7088-4133-a8a9-6dab8d37001c¹depends_on_disabled_cells§runtimeÎû!µpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$f8ccc269-0ea0-4601-ab4b-04b6aa1526ecŠ¦queued¤logs§running¦output†¤bodyÚ–

(왜?)

행렬 ${\bf A}$는 실대칭행렬ì´ë¯€ë¡œ ì§êµëŒ€ê°í™”ê°€ 가능하다. ì¼ë‹¨ 서로 ì§êµí•˜ëŠ” $n$ê°œì˜ ê³ ìœ ë²¡í„° $\psi_1,\dots \psi_n$ì„ í™•ë³´í•  수 있다. $\psi_1$ì— ëŒ€í•˜ì—¬

$$\psi_1^\top {\bf A}\psi_1=\psi_1^\top \lambda_1\psi_1=\lambda_1>0$$

ê°€ 성립한다. 여기ì—ì„œ 첫번째 등호는 ê³ ìœ ë²¡í„°ì˜ ì •ì˜, ë‘번째 등호는 $\Psi$ê°€ ì§êµí–‰ë ¬ì´ë¼ëŠ” 사실, ë§ˆì§€ë§‰ì˜ ë¶€ë“±í˜¸ëŠ” positive definiteì˜ ì •ì˜ì— ì˜í•˜ì—¬ 성립한다. ë”°ë¼ì„œ $\lambda_1>0$ì´ë‹¤. ì´ëŸ¬í•œ ë…¼ì˜ê°€ $\psi_2,\dots \psi_n$ì— ëŒ€í•˜ì—¬ 성립하므로 모든 ê³ ìœ ê°’ì€ ì–‘ìˆ˜ê°€ ëœë‹¤.

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Instantiating... === Resolving... ===  No Changes to `/tmp/jl_wbC3yq/Project.toml`  Updating `/tmp/jl_wbC3yq/Manifest.toml`  [7b1f6079] + FileWatching Precompiling... ===  Activating project at `/tmp/jl_wbC3yq`­LinearAlgebraÚm Instantiating... === Resolving... ===  No Changes to `/tmp/jl_wbC3yq/Project.toml`  Updating `/tmp/jl_wbC3yq/Manifest.toml`  [7b1f6079] + FileWatching Precompiling... ===  Activating project at `/tmp/jl_wbC3yq`§PlutoUIÚm Instantiating... === Resolving... ===  No Changes to `/tmp/jl_wbC3yq/Project.toml`  Updating `/tmp/jl_wbC3yq/Manifest.toml`  [7b1f6079] + FileWatching Precompiling... ===  Activating project at `/tmp/jl_wbC3yq`§enabled÷restart_recommended_msgÀ´restart_required_msgÀ­busy_packages¶waiting_for_permissionÂÙ,waiting_for_permission_but_probably_disabled«cell_inputsÞ'Ù$43a2c50f-0427-4c2c-af3e-c83d92c8e82c„§cell_idÙ$43a2c50f-0427-4c2c-af3e-c83d92c8e82c¤codeÚmmd""" `-` ì´ë¡ : 모든 ì›ì†Œê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ (1) 대ê°í™”가능하고 (2) ì‹¤ìˆ˜ì˜ ê³ ìœ ê°’ì„ ê°€ì§€ë©° (3) ê³ ìœ ë²¡í„°í–‰ë ¬ì´ ì§êµí–‰ë ¬ì´ë‹¤. 즉 모든 ì›ì†Œê°€ 실대칭행렬 ${\bf A}$는 아래와 ê°™ì´ í‘œí˜„í•  수 있다. ${\bf A} = {\bf \Psi}{\bf \Lambda}{\bf \Psi}^\top$ 단 ${\bf \Lambda}$ì˜ ëª¨ë“  대ê°ì„  ì›ì†ŒëŠ” 실수ì´ë‹¤. - note: ì´ ì •ë¦¬ë¥¼ 스펙트럼정리 (spectral theorem) í˜¹ì€ ì£¼ì¶•ì •ë¦¬ (principal axis theorem) ì´ë¼ê³  한다. - note: ì´ëŸ¬í•œ 매트릭스 ${\bf A}$를 ì§êµëŒ€ê°í™”가능 (orthogonally diagonalizable) ì´ë¼ê³  부른다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$99edfadb-a471-4952-90f5-ba2464eaea95„§cell_idÙ$99edfadb-a471-4952-90f5-ba2464eaea95¤codeÙ>let A = [1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1] eigen(A) end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$c226e8e1-e97c-4039-8baa-f82a00254917„§cell_idÙ$c226e8e1-e97c-4039-8baa-f82a00254917¤codeÙ.md""" ### Positive definite matrix ì •ì˜ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$0b882110-64b9-49c7-8ffd-66b95f0b8a5d„§cell_idÙ$0b882110-64b9-49c7-8ffd-66b95f0b8a5d¤codeÙ>let A = [-33 1 1 -0.22] λ,Ψ = eigen(A) # Ψ*Ψ' end ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$031acc24-0027-4db0-9270-b63a026e9575„§cell_idÙ$031acc24-0027-4db0-9270-b63a026e9575¤codeÚMmd""" `-` ì •ì˜: ì–´ë– í•œ 매트릭스 ${\bf A}_{n\times n}$ê°€ 모든 non-zero vector ${\bf y}_{n \times 1}$ì— ëŒ€í•˜ì—¬ $${\bf y}^\top {\bf A}{\bf y} > 0$$ ì„ ë§Œì¡±í•˜ë©´ ${\bf A}$를 positive definite matrixë¼ê³  부른다. - note: 참고로 등호가 있는 경우는 positive semidefinite matrixë¼ê³  부른다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$df004a39-17f0-47eb-9725-d46dd844f205„§cell_idÙ$df004a39-17f0-47eb-9725-d46dd844f205¤codeÙ,@bind y2 Slider(-50:0.1:50, show_value=true)¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$4eabb492-2f64-4eb1-9041-a8d29eb40ff8„§cell_idÙ$4eabb492-2f64-4eb1-9041-a8d29eb40ff8¤codeÙ+let A = [2 0; 2 2] y=[y1,y2] y'*A*y end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$eac996bd-5458-40eb-a82d-88f3aac35033„§cell_idÙ$eac996bd-5458-40eb-a82d-88f3aac35033¤codeÙ‰md""" (예제1) 예를들면 ${\bf A}=\begin{bmatrix} 2 & 0 \\ 2 & 2 \end{bmatrix}$는 positive definite ì˜ ì •ì˜ë¥¼ 만족한다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$1d6ec17b-eca2-4289-8cfe-04a67bcac2c6„§cell_idÙ$1d6ec17b-eca2-4289-8cfe-04a67bcac2c6¤codeÚzhtml"""
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$94b5adf6-cbe5-4052-a2bd-dfabac6d7406„§cell_idÙ$94b5adf6-cbe5-4052-a2bd-dfabac6d7406¤codeÙ}md""" (예제2) ${\bf A}=\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$ 는 positive definiteì˜ ì •ì˜ë¥¼ 만족한다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$6cc96118-41ac-4807-8f88-4ae86e491c79„§cell_idÙ$6cc96118-41ac-4807-8f88-4ae86e491c79¤codeÙ9let A = [1 1 -1 1] y=[y1,y2] y'*A*y #eigen(A) end ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$9022b592-6ab0-40de-b96b-aecb56c663a5„§cell_idÙ$9022b592-6ab0-40de-b96b-aecb56c663a5¤codeÙ\let ψ1= 1/√2 .* [1,-im] ψ2= 1/√2 .* [im,1] Ψ = [ψ1 ψ2] rank(Ψ) #inv(Ψ) end ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$bd476cf8-9946-4911-bee9-5170aabd9ca7„§cell_idÙ$bd476cf8-9946-4911-bee9-5170aabd9ca7¤codeÙ€md""" (예제3) 대칭행렬ì´ì§€ë§Œ 모든 ì›ì†Œê°€ 실수가 ì•„ë‹Œ 경우ì—는 대ê°í™” 불가능 í•  ìˆ˜ë„ ìžˆë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$350200af-c763-4bc3-8b7f-be8422988009„§cell_idÙ$350200af-c763-4bc3-8b7f-be8422988009¤codeÙsmd""" > ì´ì „ì¦ëª…ì„ ì‚­ì œí•˜ê³  ì´ ì¦ëª…으로 수정하였습니다. ì´ ì¦ëª…으로 공부하세요!! """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$2430ceb6-1154-4345-8487-05bd9a8136e2„§cell_idÙ$2430ceb6-1154-4345-8487-05bd9a8136e2¤codeÙ:let A = [2im 1 1 0] λ,Ψ = eigen(A) #rank(Ψ) end ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$05af5915-30a6-49fd-850a-352a41036984„§cell_idÙ$05af5915-30a6-49fd-850a-352a41036984¤codeÚUmd""" - ì´ë•Œ $\lambda=i$ ì´ê³  $\psi_1=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -i \end{bmatrix}$, $\psi_2 = \frac{1}{\sqrt{2}} \begin{bmatrix} i \\ 1 \end{bmatrix}$ ì´ë‹¤. - $\psi_1 i = \psi_2$ìž„ì„ ê´€ì°°í•˜ë¼. (ë”°ë¼ì„œ ê³ ìœ ë²¡í„°í–‰ë ¬ì´ full rankê°€ 아니다) - 즉 ì´ í–‰ë ¬ì€ ëŒ€ê°í™”ê°€ 불가능한 행렬ì´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$4d737613-177c-45e8-abe7-597ffd863049„§cell_idÙ$4d737613-177c-45e8-abe7-597ffd863049¤codeÙ,@bind y1 Slider(-50:0.1:50, show_value=true)¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$d3ccc0c6-e092-11ec-3296-b5a32d51eade„§cell_idÙ$d3ccc0c6-e092-11ec-3296-b5a32d51eade¤code³md""" ## usings """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$12aa1c0d-8f14-48cf-b244-c3fac884d676„§cell_idÙ$12aa1c0d-8f14-48cf-b244-c3fac884d676¤code½md""" ## Spectral theorem """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$7befd7ef-7ab9-4a00-a9d8-52cde80c23e6„§cell_idÙ$7befd7ef-7ab9-4a00-a9d8-52cde80c23e6¤code±md""" (왜?) """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$742c33c4-3bd5-4388-a9dd-4f9341eeb3e4„§cell_idÙ$742c33c4-3bd5-4388-a9dd-4f9341eeb3e4¤codeÙymd""" (예제1) 0í–‰ë ¬ì€ ëª¨ë“  ì›ì†Œê°€ 실수ì´ë©° 대칭행렬ì´ë‹¤. ë”°ë¼ì„œ ì§êµëŒ€ê°í™” 가능하다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$9b52b105-40b4-4b77-b274-2d0a6914985a„§cell_idÙ$9b52b105-40b4-4b77-b274-2d0a6914985a¤codeÙ$let A = [0 0 ; 0 0] eigen(A) end ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$fb480817-66f5-468b-b6c6-9bf145fc86c4„§cell_idÙ$fb480817-66f5-468b-b6c6-9bf145fc86c4¤codeÙ4md""" ### Positive definite and symmetric matrix """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$27d01165-2aa4-4b86-93ba-c7109636e692„§cell_idÙ$27d01165-2aa4-4b86-93ba-c7109636e692¤codeµmd""" # 5ì›”31ì¼ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$b44292d9-2a3f-49d6-bcc2-8fb5b72e9e6c„§cell_idÙ$b44292d9-2a3f-49d6-bcc2-8fb5b72e9e6c¤codeÙÈmd""" `-` 요약: ${\bf A}$ê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ì´ë©´ 아래가 ë™ì¹˜ì´ë‹¤. - 행렬 ${\bf A}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì´ë‹¤. - 행렬 ${\bf A}$ê°€ positive definite matrixì´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$dc456627-72c5-4acf-89af-aa1ba01d2ba4„§cell_idÙ$dc456627-72c5-4acf-89af-aa1ba01d2ba4¤codeÙYmd""" - note: 참고로 ì´ë•Œ ${\bf A}$ì˜ ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì¸ ê²ƒì€ ì•„ë‹ˆë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$dc1ad9d6-98e3-4b1c-a90b-5f6a6a60806f„§cell_idÙ$dc1ad9d6-98e3-4b1c-a90b-5f6a6a60806f¤codeÙ(3im + 1 # 3i + 1 ê³¼ 같다. 즉, 허수¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$da1a6561-ef07-492c-8e20-bfb0dc530a05„§cell_idÙ$da1a6561-ef07-492c-8e20-bfb0dc530a05¤codeÙÓmd""" `-` 요약: ${\bf A}$ê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ì´ë©´ 아래가 ë™ì¹˜ì´ë‹¤. - 행렬 ${\bf A}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì´ 0ë˜ëŠ” 양수ì´ë‹¤. - 행렬 ${\bf A}$ê°€ positive semidefinite matrixì´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$f080d7b0-4c1f-411e-9e86-d422dd77d2e7„§cell_idÙ$f080d7b0-4c1f-411e-9e86-d422dd77d2e7¤codeÙ%md""" ## Positive definite matrix """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$a23df07e-24df-4ae3-942d-7ccfe024de57„§cell_idÙ$a23df07e-24df-4ae3-942d-7ccfe024de57¤codeÙ›md""" ## 숙제 ê³ ìœ ë²¡í„°í–‰ë ¬ì´ ì§êµí–‰ë ¬ì´ê³  모든 ê³ ìœ ê°’ì´ ì‹¤ìˆ˜ê°€ ë˜ëŠ” 매트릭스 ${\bf A}_{4\times 4}$ 를 구해보ë¼. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$26300fa0-9f73-4874-9e60-d7b82f05cc71„§cell_idÙ$26300fa0-9f73-4874-9e60-d7b82f05cc71¤code¹PlutoUI.TableOfContents()¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$5b94d293-7533-4f4d-8c71-0a2c19e94117„§cell_idÙ$5b94d293-7533-4f4d-8c71-0a2c19e94117¤codeÙEmd""" (why?) ${\bf y}'{\bf A}{\bf y}=(y_1+y_2)^2+y_1^2+y_2^2>0$ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$3e069b39-28fb-4404-b292-8a54c2a25a8a„§cell_idÙ$3e069b39-28fb-4404-b292-8a54c2a25a8a¤codeÚmd""" `-` 관찰1: ì–´ë– í•œ 실대칭행렬 ${\bf A}_{n\times n}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì´ë©´ ${\bf A}$는 positive definite matrixê°€ ëœë‹¤. (즉 모든 non-zero vector ${\bf y}$ì— ëŒ€í•´ì„œ ${\bf y}^\top {\bf A}{\bf y}>0$ì„ ë§Œì¡±í•œë‹¤.) """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$71d85e9f-a649-4584-8cbd-d43dedba97cb„§cell_idÙ$71d85e9f-a649-4584-8cbd-d43dedba97cb¤codeÚ#md""" 모든 ìž„ì˜ì˜ ${\bf y}$ì— ëŒ€í•˜ì—¬ 아래가 성리하므로. $${\bf y}^\top{\bf A}{\bf y}={\bf y}^\top{\bf \Psi}{\bf \Lambda}{\bf \Psi}^\top{\bf y}={\bf x}^\top {\bf \Lambda}{\bf x}=\sum_{i=1}^{n}x_i^2\lambda_i>0$$ 단 여기ì—ì„œ ${\bf x}={\bf \Psi}^\top{\bf y}$ ì´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$f1c60bf5-2052-4bf3-8e3b-ddb0cbd14b39„§cell_idÙ$f1c60bf5-2052-4bf3-8e3b-ddb0cbd14b39¤codeÙqmd""" (예제2) 아무렇게나 ì‹¤ìˆ˜ì¸ ëŒ€ì¹­í–‰ë ¬ì„ ë§Œë“¤ê¸°ë§Œí•˜ë©´ ì§êµëŒ€ê°í™”ê°€ 가능하다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$b868981c-dffb-4165-8e69-97889bc25312„§cell_idÙ$b868981c-dffb-4165-8e69-97889bc25312¤code»using LinearAlgebra,PlutoUI¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$706c2e23-626e-44e0-a443-70023ffbccc6„§cell_idÙ$706c2e23-626e-44e0-a443-70023ffbccc6¤codeÙômd""" `-` 관찰2: ì–´ë– í•œ 실대칭행렬 ${\bf A}$ê°€ positive definite matrix ì´ë©´ (즉 모든 non-zero vector ${\bf y}$ì— ëŒ€í•˜ì—¬ ${\bf y}^\top {\bf A}{\bf y}>0$ì´ ì„±ë¦½í•œë‹¤ë©´) ${\bf A}$ì˜ ëª¨ë“  ê³ ìœ ê°’ì€ ì–‘ìˆ˜ì´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$1bcbd243-7088-4133-a8a9-6dab8d37001c„§cell_idÙ$1bcbd243-7088-4133-a8a9-6dab8d37001c¤codeÙKmd""" - note: 참고로 ì´ë•Œ ${\bf A}$는 ëŒ€ì¹­í–‰ë ¬ì´ ì•„ë‹ˆë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$f8ccc269-0ea0-4601-ab4b-04b6aa1526ec„§cell_idÙ$f8ccc269-0ea0-4601-ab4b-04b6aa1526ec¤codeÚƒmd""" (왜?) 행렬 ${\bf A}$는 실대칭행렬ì´ë¯€ë¡œ ì§êµëŒ€ê°í™”ê°€ 가능하다. ì¼ë‹¨ 서로 ì§êµí•˜ëŠ” $n$ê°œì˜ ê³ ìœ ë²¡í„° $\psi_1,\dots \psi_n$ì„ í™•ë³´í•  수 있다. $\psi_1$ì— ëŒ€í•˜ì—¬ $\psi_1^\top {\bf A}\psi_1=\psi_1^\top \lambda_1\psi_1=\lambda_1>0$ ê°€ 성립한다. 여기ì—ì„œ 첫번째 등호는 ê³ ìœ ë²¡í„°ì˜ ì •ì˜, ë‘번째 등호는 $\Psi$ê°€ ì§êµí–‰ë ¬ì´ë¼ëŠ” 사실, ë§ˆì§€ë§‰ì˜ ë¶€ë“±í˜¸ëŠ” positive definiteì˜ ì •ì˜ì— ì˜í•˜ì—¬ 성립한다. ë”°ë¼ì„œ $\lambda_1>0$ì´ë‹¤. ì´ëŸ¬í•œ ë…¼ì˜ê°€ $\psi_2,\dots \psi_n$ì— ëŒ€í•˜ì—¬ 성립하므로 모든 ê³ ìœ ê°’ì€ ì–‘ìˆ˜ê°€ ëœë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedënotebook_idÙ$338c3df6-e472-11ef-382f-5fd6b23d3a27«in_temp_dir¨metadata€