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(ì˜ˆì œ2) ${\bf A}=\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$ 는 positive definiteì˜ ì •ì˜ë¥¼ 만족한다.
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(ì˜ˆì œ3) 대ì¹í–‰ë ¬ì´ì§€ë§Œ ëª¨ë“ 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 ìˆ˜ì •í•˜ì˜€ìŠµë‹ˆë‹¤. ì´ ì¦ëª…으로 공부하세요!!
¤mime©text/html¬rootassigneeÀ²last_run_timestampËAÙ09§Ð-°persist_js_state·has_pluto_hook_features§cell_idÙ$350200af-c763-4bc3-8b7f-be8422988009¹depends_on_disabled_cells§runtimeÎ'ÁZµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$2430ceb6-1154-4345-8487-05bd9a8136e2Š¦queued¤logs§running¦output†¤bodyÚ@Eigen{ComplexF64, ComplexF64, Matrix{ComplexF64}, Vector{ComplexF64}}
values:
2-element Vector{ComplexF64}:
-1.4686563106965222e-8 + 1.0000000025198155im
1.4686562910259937e-8 + 0.9999999974801843im
vectors:
2×2 Matrix{ComplexF64}:
0.707107+0.0im 1.0385e-8+0.707107im
-1.0385e-8-0.707107im 0.707107+0.0im¤mimeªtext/plain¬rootassigneeÀ²last_run_timestampËAÙ094ÕË°persist_js_state·has_pluto_hook_features§cell_idÙ$2430ceb6-1154-4345-8487-05bd9a8136e2¹depends_on_disabled_cells§runtimeÎù-Ûµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$05af5915-30a6-49fd-850a-352a41036984Š¦queued¤logs§running¦output†¤bodyÚ
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 í–‰ë ¬ì´ë‹¤.
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usings
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Spectral theorem
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ì´ ê°•ì˜ë…¸íŠ¸ì˜ PD-matrix는 ì¼ë¶€ë¬¸í—Œì˜ ì •ì˜ë¥¼ 사용하여 설명한 것입니다. (예시는 아래..)
- Golub, G. H., & Van Loan, C. (1979). Unsymmetric positive definite linear systems. Linear Algebra and its Applications, 28, 85-97.
현재는 PD-matrixê°€ 실대ì¹í–‰ë ¬(or ì—ë¥´ë¯¸íŠ¸í–‰ë ¬)ìž„ì„ ì „ìž¬í•˜ê³ ì„œìˆ í•œ ë¬¸í˜„ì´ ë§Žìœ¼ë¯€ë¡œ 해당내용으로 공부하시기 ë°”ëžë‹ˆë‹¤. 2024ë…„ ê°•ì˜ë…¸íŠ¸ ë²„ì „ <https://github.com/guebin/SC2024> ì—서는 다른 ì •ì˜ë¡œ PD를 설명합니다.
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(ì˜ˆì œ1) 0í–‰ë ¬ì€ ëª¨ë“ ì›ì†Œê°€ 실수ì´ë©° 대ì¹í–‰ë ¬ì´ë‹¤. ë”°ë¼ì„œ ì§êµëŒ€ê°í™” 가능하다.
¤mime©text/html¬rootassigneeÀ²last_run_timestampËAÙ08ƒ¥Ê°persist_js_state·has_pluto_hook_features§cell_idÙ$742c33c4-3bd5-4388-a9dd-4f9341eeb3e4¹depends_on_disabled_cells§runtimeÎ _ʵpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$9b52b105-40b4-4b77-b274-2d0a6914985aŠ¦queued¤logs§running¦output†¤bodyÙ™Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
2-element Vector{Float64}:
0.0
0.0
vectors:
2×2 Matrix{Float64}:
1.0 0.0
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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‹¤.
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- 요약: ${\bf A}$ê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹í–‰ë ¬ì´ë©´ 아래가 ë™ì¹˜ì´ë‹¤.
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Positive definite matrix
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ìˆ™ì œ
ê³ ìœ ë²¡í„°í–‰ë ¬ì´ ì§êµí–‰ë ¬ì´ê³ ëª¨ë“ ê³ ìœ ê°’ì´ ì‹¤ìˆ˜ê°€ ë˜ëŠ” 매트ë¦ìŠ¤ ${\bf A}_{4\times 4}$ 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 실대ì¹í–‰ë ¬ ${\bf A}_{n\times n}$ì˜ ëª¨ë“ ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì´ë©´ ${\bf A}$는 positive definite matrixê°€ ëœë‹¤. (즉 ëª¨ë“ 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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ëª¨ë“ 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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(ì˜ˆì œ2) ì•„ë¬´ë ‡ê²Œë‚˜ ì‹¤ìˆ˜ì¸ ëŒ€ì¹í–‰ë ¬ì„ 만들기만하면 ì§êµëŒ€ê°í™”ê°€ 가능하다.
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- 관찰2: ì–´ë– í•œ 실대ì¹í–‰ë ¬ ${\bf A}$ê°€ positive definite matrix ì´ë©´ (즉 ëª¨ë“ 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‹¤.
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(왜?)
í–‰ë ¬ ${\bf A}$는 실대ì¹í–‰ë ¬ì´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• ìˆ˜ 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 ëª¨ë“ ê³ ìœ ê°’ì€ ì–‘ìˆ˜ê°€ ëœë‹¤.
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[90m [8e850ede] [39m[92m+ nghttp2_jll[39m
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6 dependencies successfully precompiled in 6 seconds (15 already precompiled)LinearAlgebraÚ
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6 dependencies successfully precompiled in 6 seconds (15 already precompiled)§PlutoUIÚ
Instantiating...
[32m[1m Installed[22m[39m FixedPointNumbers ────── v0.8.4
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[32m[1m Installed[22m[39m ColorTypes ───────────── v0.11.3
[32m[1m Installed[22m[39m IOCapture ────────────── v0.2.2
[32m[1m Installed[22m[39m Tricks ───────────────── v0.1.6
[32m[1m Installed[22m[39m PlutoUI ──────────────── v0.7.39
[32m[1m Installed[22m[39m JSON ─────────────────── v0.21.3
[32m[1m Installed[22m[39m Reexport ─────────────── v1.2.2
Installed HypertextLiteral ─────── v0.9.4 [32m[1m[22m[39m
Installed AbstractPlutoDingetjes ─ v1.1.4 [32m[1m[22m[39m
Resolving...
[36m[1m┌ [22m[39m[36m[1mInfo: [22m[39mOperation failed. Updating registries and trying again...
[36m[1m│ [22m[39m exception =
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[36m[1m│ [22m[39m MozillaCACerts_jll [14a3606d] log:
[36m[1m│ [22m[39m ├─possible versions are: 2020.7.22 or uninstalled
[36m[1m└ [22m[39m └─restricted to versions 2022.2.1 by an explicit requirement — no versions left
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[33m[1m┌ [22m[39m[33m[1mWarning: [22m[39mOperation failed. Removing Manifest and trying again...
[33m[1m│ [22m[39m exception =
[33m[1m│ [22m[39m Unsatisfiable requirements detected for package MozillaCACerts_jll [14a3606d]:
[33m[1m│ [22m[39m MozillaCACerts_jll [14a3606d] log:
[33m[1m│ [22m[39m ├─possible versions are: 2020.7.22 or uninstalled
[33m[1m│ [22m[39m └─restricted to versions 2022.2.1 by an explicit requirement — no versions left
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Resolving...
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[32m[1m Installed[22m[39m PlutoUI ───── v0.7.59
Installed Hyperscript ─ v0.0.5 [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_GPgAC0/Project.toml`
[90m [7f904dfe] [39m[92m+ PlutoUI v0.7.59[39m
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[90m [53c48c17] [39m[92m+ FixedPointNumbers v0.8.4[39m
[90m [47d2ed2b] [39m[92m+ Hyperscript v0.0.5[39m
[90m [ac1192a8] [39m[92m+ HypertextLiteral v0.9.5[39m
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[90m [682c06a0] [39m[92m+ JSON v0.21.4[39m
[90m [6c6e2e6c] [39m[92m+ MIMEs v0.1.4[39m
[90m [69de0a69] [39m[92m+ Parsers v2.8.1[39m
[90m [7f904dfe] [39m[92m+ PlutoUI v0.7.59[39m
[90m [aea7be01] [39m[92m+ PrecompileTools v1.2.1[39m
[90m [21216c6a] [39m[92m+ Preferences v1.4.3[39m
[90m [189a3867] [39m[92m+ Reexport v1.2.2[39m
[90m [410a4b4d] [39m[92m+ Tricks v0.1.8[39m
[90m [5c2747f8] [39m[92m+ URIs v1.5.1[39m
[90m [0dad84c5] [39m[92m+ ArgTools[39m
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[90m [f43a241f] [39m[92m+ Downloads[39m
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[90m [b77e0a4c] [39m[92m+ InteractiveUtils[39m
[90m [b27032c2] [39m[92m+ LibCURL[39m
[90m [76f85450] [39m[92m+ LibGit2[39m
[90m [8f399da3] [39m[92m+ Libdl[39m
[90m [37e2e46d] [39m[92m+ LinearAlgebra[39m
[90m [56ddb016] [39m[92m+ Logging[39m
[90m [d6f4376e] [39m[92m+ Markdown[39m
[90m [a63ad114] [39m[92m+ Mmap[39m
[90m [ca575930] [39m[92m+ NetworkOptions[39m
[90m [44cfe95a] [39m[92m+ Pkg[39m
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[90m [9a3f8284] [39m[92m+ Random[39m
[90m [ea8e919c] [39m[92m+ SHA[39m
[90m [9e88b42a] [39m[92m+ Serialization[39m
[90m [6462fe0b] [39m[92m+ Sockets[39m
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[90m [10745b16] [39m[92m+ Statistics[39m
[90m [fa267f1f] [39m[92m+ TOML[39m
[90m [a4e569a6] [39m[92m+ Tar[39m
[90m [8dfed614] [39m[92m+ Test[39m
[90m [cf7118a7] [39m[92m+ UUIDs[39m
[90m [4ec0a83e] [39m[92m+ Unicode[39m
[90m [e66e0078] [39m[92m+ CompilerSupportLibraries_jll[39m
[90m [deac9b47] [39m[92m+ LibCURL_jll[39m
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[90m [c8ffd9c3] [39m[92m+ MbedTLS_jll[39m
[90m [14a3606d] [39m[92m+ MozillaCACerts_jll[39m
[90m [4536629a] [39m[92m+ OpenBLAS_jll[39m
[90m [83775a58] [39m[92m+ Zlib_jll[39m
[90m [8e850b90] [39m[92m+ libblastrampoline_jll[39m
[90m [8e850ede] [39m[92m+ nghttp2_jll[39m
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6 dependencies successfully precompiled in 6 seconds (15 already precompiled)§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¡ : ëª¨ë“ ì›ì†Œê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹í–‰ë ¬ (1) 대ê°í™”ê°€ëŠ¥í•˜ê³ (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 ëª¨ë“ ì›ì†Œê°€ 실대ì¹í–‰ë ¬ ${\bf A}$는 아래와 ê°™ì´ í‘œí˜„í• ìˆ˜ 있다.
${\bf A} = {\bf \Psi}{\bf \Lambda}{\bf \Psi}^\top$
단 ${\bf \Lambda}$ì˜ ëª¨ë“ ëŒ€ê°ì„ ì›ì†ŒëŠ” 실수ì´ë‹¤.
- note: ì´ ì •ë¦¬ë¥¼ ìŠ¤íŽ™íŠ¸ëŸ¼ì •ë¦¬ (spectral theorem) í˜¹ì€ ì£¼ì¶•ì •ë¦¬ (principal axis theorem) ì´ë¼ê³ 한다.
- note: ì´ëŸ¬í•œ 매트ë¦ìŠ¤ ${\bf A}$를 ì§êµëŒ€ê°í™”가능 (orthogonally diagonalizable) ì´ë¼ê³ 부른다.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$c226e8e1-e97c-4039-8baa-f82a00254917„§cell_idÙ$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¤codeÚUmd"""
`-` ì •ì˜: ì–´ë– í•œ 실대ì¹í–‰ë ¬ ${\bf A}_{n\times n}$ê°€ ëª¨ë“ non-zero real 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: ì°¸ê³ ë¡œ 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"""
(ì˜ˆì œ1) 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_foldedÂÙ$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"""
(ì˜ˆì œ2) ${\bf A}=\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$ 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"""
(ì˜ˆì œ3) 대ì¹í–‰ë ¬ì´ì§€ë§Œ ëª¨ë“ 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 ìˆ˜ì •í•˜ì˜€ìŠµë‹ˆë‹¤. 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 í–‰ë ¬ì´ë‹¤.
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## 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$1309cb46-7027-45b4-9401-0ed38a1ef7a7„§cell_idÙ$1309cb46-7027-45b4-9401-0ed38a1ef7a7¤codeÚ md"""
ì´ ê°•ì˜ë…¸íŠ¸ì˜ PD-matrix는 ì¼ë¶€ë¬¸í—Œì˜ ì •ì˜ë¥¼ 사용하여 설명한 것입니다. (예시는 아래..)
- Golub, G. H., & Van Loan, C. (1979). Unsymmetric positive definite linear systems. Linear Algebra and its Applications, 28, 85-97.
현재는 PD-matrixê°€ 실대ì¹í–‰ë ¬(or ì—ë¥´ë¯¸íŠ¸í–‰ë ¬)ìž„ì„ ì „ìž¬í•˜ê³ ì„œìˆ í•œ ë¬¸í˜„ì´ ë§Žìœ¼ë¯€ë¡œ 해당내용으로 공부하시기 ë°”ëžë‹ˆë‹¤. 2024ë…„ ê°•ì˜ë…¸íŠ¸ ë²„ì „ ì—서는 다른 ì •ì˜ë¡œ PD를 설명합니다.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$742c33c4-3bd5-4388-a9dd-4f9341eeb3e4„§cell_idÙ$742c33c4-3bd5-4388-a9dd-4f9341eeb3e4¤codeÙymd"""
(ì˜ˆì œ1) 0í–‰ë ¬ì€ ëª¨ë“ ì›ì†Œê°€ 실수ì´ë©° 대ì¹í–‰ë ¬ì´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$7befd7ef-7ab9-4a00-a9d8-52cde80c23e6„§cell_idÙ$7befd7ef-7ab9-4a00-a9d8-52cde80c23e6¤code±md"""
(왜?)
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$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‹¤.
- í–‰ë ¬ ${\bf A}$ì˜ ëª¨ë“ ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì´ë‹¤.
- í–‰ë ¬ ${\bf A}$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: ì°¸ê³ ë¡œ 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$da1a6561-ef07-492c-8e20-bfb0dc530a05„§cell_idÙ$da1a6561-ef07-492c-8e20-bfb0dc530a05¤codeÙÓmd"""
`-` 요약: ${\bf A}$ê°€ ì‹¤ìˆ˜ì¸ ëŒ€ì¹í–‰ë ¬ì´ë©´ 아래가 ë™ì¹˜ì´ë‹¤.
- í–‰ë ¬ ${\bf A}$ì˜ ëª¨ë“ ê³ ìœ ê°’ì´ 0ë˜ëŠ” 양수ì´ë‹¤.
- í–‰ë ¬ ${\bf A}$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"""
## ìˆ™ì œ
ê³ ìœ ë²¡í„°í–‰ë ¬ì´ ì§êµí–‰ë ¬ì´ê³ ëª¨ë“ ê³ ìœ ê°’ì´ ì‹¤ìˆ˜ê°€ ë˜ëŠ” 매트ë¦ìŠ¤ ${\bf A}_{4\times 4}$ 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¤codeÚ md"""
`-` 관찰1: ì–´ë– í•œ 실대ì¹í–‰ë ¬ ${\bf A}_{n\times n}$ì˜ ëª¨ë“ ê³ ìœ ê°’ì´ ì–‘ìˆ˜ì´ë©´ ${\bf A}$는 positive definite matrixê°€ ëœë‹¤. (즉 ëª¨ë“ 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"""
ëª¨ë“ 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"""
(ì˜ˆì œ2) ì•„ë¬´ë ‡ê²Œë‚˜ ì‹¤ìˆ˜ì¸ ëŒ€ì¹í–‰ë ¬ì„ 만들기만하면 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 실대ì¹í–‰ë ¬ ${\bf A}$ê°€ positive definite matrix ì´ë©´ (즉 ëª¨ë“ 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: ì°¸ê³ ë¡œ ì´ë•Œ ${\bf A}$는 대ì¹í–‰ë ¬ì´ 아니다.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$f8ccc269-0ea0-4601-ab4b-04b6aa1526ec„§cell_idÙ$f8ccc269-0ea0-4601-ab4b-04b6aa1526ec¤codeÚƒmd"""
(왜?)
í–‰ë ¬ ${\bf A}$는 실대ì¹í–‰ë ¬ì´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• ìˆ˜ 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 ëª¨ë“ ê³ ìœ ê°’ì€ ì–‘ìˆ˜ê°€ ëœë‹¤.
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