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[32m[1m No Changes[22m[39m to `/tmp/jl_Z2B6Vl/Project.toml` [32m[1m Updating[22m[39m `/tmp/jl_Z2B6Vl/Manifest.toml` [90m [6e34b625] [39m[93m↑ Bzip2_jll v1.0.8+0 ⇒ v1.0.8+1[39m [90m [61579ee1] [39m[93m↑ Ghostscript_jll v9.55.0+0 ⇒ v9.55.0+4[39m [90m [c73af94c] [39m[93m↑ ImageMagick_jll v6.9.12+2 ⇒ v6.9.12+4[39m [90m [94ce4f54] [39m[93m↑ Libiconv_jll v1.16.1+1 ⇒ v1.16.1+2[39m [90m [d3a379c0] [39m[92m+ LittleCMS_jll v2.12.0+0[39m [90m [643b3616] [39m[92m+ OpenJpeg_jll v2.4.0+0[39m [90m [ea2cea3b] [39m[93m↑ Qt5Base_jll v5.15.3+1 ⇒ v5.15.3+2[39m [90m [b53b4c65] [39m[93m↑ libpng_jll v1.6.38+0 ⇒ v1.6.38+1[39m [90m [7b1f6079] [39m[92m+ FileWatching[39m Precompiling... [32m[1m Activating[22m[39m project at `/tmp/jl_Z2B6Vl`©RDatasetsÚJ Instantiating... Resolving... [32m[1m No Changes[22m[39m to `/tmp/jl_Z2B6Vl/Project.toml` [32m[1m Updating[22m[39m `/tmp/jl_Z2B6Vl/Manifest.toml` [90m [6e34b625] [39m[93m↑ Bzip2_jll v1.0.8+0 ⇒ v1.0.8+1[39m [90m [61579ee1] [39m[93m↑ Ghostscript_jll v9.55.0+0 ⇒ v9.55.0+4[39m [90m [c73af94c] [39m[93m↑ ImageMagick_jll v6.9.12+2 ⇒ v6.9.12+4[39m [90m [94ce4f54] [39m[93m↑ Libiconv_jll v1.16.1+1 ⇒ v1.16.1+2[39m [90m [d3a379c0] [39m[92m+ LittleCMS_jll v2.12.0+0[39m [90m [643b3616] [39m[92m+ OpenJpeg_jll v2.4.0+0[39m [90m [ea2cea3b] [39m[93m↑ Qt5Base_jll v5.15.3+1 ⇒ v5.15.3+2[39m [90m [b53b4c65] [39m[93m↑ libpng_jll v1.6.38+0 ⇒ v1.6.38+1[39m [90m [7b1f6079] [39m[92m+ FileWatching[39m Precompiling... [32m[1m Activating[22m[39m project at `/tmp/jl_Z2B6Vl`LinearAlgebraÚJ Instantiating... Resolving... [32m[1m No Changes[22m[39m to `/tmp/jl_Z2B6Vl/Project.toml` [32m[1m Updating[22m[39m `/tmp/jl_Z2B6Vl/Manifest.toml` [90m [6e34b625] [39m[93m↑ Bzip2_jll v1.0.8+0 ⇒ v1.0.8+1[39m [90m [61579ee1] [39m[93m↑ Ghostscript_jll v9.55.0+0 ⇒ v9.55.0+4[39m [90m [c73af94c] [39m[93m↑ ImageMagick_jll v6.9.12+2 ⇒ v6.9.12+4[39m [90m [94ce4f54] [39m[93m↑ Libiconv_jll v1.16.1+1 ⇒ v1.16.1+2[39m [90m [d3a379c0] [39m[92m+ LittleCMS_jll v2.12.0+0[39m [90m [643b3616] [39m[92m+ OpenJpeg_jll v2.4.0+0[39m [90m [ea2cea3b] [39m[93m↑ Qt5Base_jll v5.15.3+1 ⇒ v5.15.3+2[39m [90m [b53b4c65] [39m[93m↑ libpng_jll v1.6.38+0 ⇒ v1.6.38+1[39m [90m [7b1f6079] [39m[92m+ FileWatching[39m Precompiling... [32m[1m Activating[22m[39m project at `/tmp/jl_Z2B6Vl`¥PlotsÚJ Instantiating... Resolving... [32m[1m No Changes[22m[39m to `/tmp/jl_Z2B6Vl/Project.toml` [32m[1m Updating[22m[39m `/tmp/jl_Z2B6Vl/Manifest.toml` [90m [6e34b625] [39m[93m↑ Bzip2_jll v1.0.8+0 ⇒ v1.0.8+1[39m [90m [61579ee1] [39m[93m↑ Ghostscript_jll v9.55.0+0 ⇒ v9.55.0+4[39m [90m [c73af94c] [39m[93m↑ ImageMagick_jll v6.9.12+2 ⇒ v6.9.12+4[39m [90m [94ce4f54] [39m[93m↑ Libiconv_jll v1.16.1+1 ⇒ v1.16.1+2[39m [90m [d3a379c0] [39m[92m+ LittleCMS_jll v2.12.0+0[39m [90m [643b3616] [39m[92m+ OpenJpeg_jll v2.4.0+0[39m [90m [ea2cea3b] [39m[93m↑ Qt5Base_jll v5.15.3+1 ⇒ v5.15.3+2[39m [90m [b53b4c65] [39m[93m↑ libpng_jll v1.6.38+0 ⇒ v1.6.38+1[39m [90m [7b1f6079] [39m[92m+ FileWatching[39m Precompiling... [32m[1m Activating[22m[39m project at `/tmp/jl_Z2B6Vl`§PlutoUIÚJ Instantiating... Resolving... [32m[1m No Changes[22m[39m to `/tmp/jl_Z2B6Vl/Project.toml` [32m[1m Updating[22m[39m `/tmp/jl_Z2B6Vl/Manifest.toml` [90m [6e34b625] [39m[93m↑ Bzip2_jll v1.0.8+0 ⇒ v1.0.8+1[39m [90m [61579ee1] [39m[93m↑ Ghostscript_jll v9.55.0+0 ⇒ v9.55.0+4[39m [90m [c73af94c] [39m[93m↑ ImageMagick_jll v6.9.12+2 ⇒ v6.9.12+4[39m [90m [94ce4f54] [39m[93m↑ Libiconv_jll v1.16.1+1 ⇒ v1.16.1+2[39m [90m [d3a379c0] [39m[92m+ LittleCMS_jll v2.12.0+0[39m [90m [643b3616] [39m[92m+ OpenJpeg_jll v2.4.0+0[39m [90m [ea2cea3b] [39m[93m↑ Qt5Base_jll v5.15.3+1 ⇒ v5.15.3+2[39m [90m [b53b4c65] [39m[93m↑ libpng_jll v1.6.38+0 ⇒ v1.6.38+1[39m [90m [7b1f6079] [39m[92m+ FileWatching[39m Precompiling... [32m[1m Activating[22m[39m project at `/tmp/jl_Z2B6Vl`¦RandomÚJ Instantiating... Resolving... [32m[1m No Changes[22m[39m to `/tmp/jl_Z2B6Vl/Project.toml` [32m[1m Updating[22m[39m `/tmp/jl_Z2B6Vl/Manifest.toml` [90m [6e34b625] [39m[93m↑ Bzip2_jll v1.0.8+0 ⇒ v1.0.8+1[39m [90m [61579ee1] [39m[93m↑ Ghostscript_jll v9.55.0+0 ⇒ v9.55.0+4[39m [90m [c73af94c] [39m[93m↑ ImageMagick_jll v6.9.12+2 ⇒ v6.9.12+4[39m [90m [94ce4f54] [39m[93m↑ Libiconv_jll v1.16.1+1 ⇒ v1.16.1+2[39m [90m [d3a379c0] [39m[92m+ LittleCMS_jll v2.12.0+0[39m [90m [643b3616] [39m[92m+ OpenJpeg_jll v2.4.0+0[39m [90m [ea2cea3b] [39m[93m↑ Qt5Base_jll v5.15.3+1 ⇒ v5.15.3+2[39m [90m [b53b4c65] [39m[93m↑ libpng_jll v1.6.38+0 ⇒ v1.6.38+1[39m [90m [7b1f6079] [39m[92m+ FileWatching[39m Precompiling... [32m[1m Activating[22m[39m project at `/tmp/jl_Z2B6Vl`§enabledìinstantiated÷restart_recommended_msgÀ´restart_required_msgÀ¯install_time_nsÎÔ_“rbusy_packages�«cell_inputsÞ VÙ$1faa1a3f-7b45-4155-afbe-708fe8e7ccaa„§cell_idÙ$1faa1a3f-7b45-4155-afbe-708fe8e7ccaa¤codeÙ0let A = [1 0 0 ; 2 3 0 ; 3 2 5] eigen(A) end ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$558aa0d2-6f92-43ca-b84b-a82430b8e0cc„§cell_idÙ$558aa0d2-6f92-43ca-b84b-a82430b8e0cc¤codeÙ~md""" (슬기) ${\bf X}$는 $150\times 4$ matrix ì�´ë¯€ë¡œ ${\bf X}$ì�˜ rank는 기ê»�해야 (=아무리 커ë�„) 4ì�´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$5e72eefe-14ea-4800-8ca7-dc89be5b5af6„§cell_idÙ$5e72eefe-14ea-4800-8ca7-dc89be5b5af6¤codeÙ\md""" (í’€ì�´) ê³ ìœ ê°’ì�˜ 곱과 대ê°�ì„ ì›�소ì�˜ ê³±ì�´ 모ë‘� 1×3×5로 같다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$af6dff08-f753-4e23-8226-314510849901„§cell_idÙ$af6dff08-f753-4e23-8226-314510849901¤codeÙamd""" (8) 중복ë�œ ê³ ìœ ê°’ì�„ ê°€ì§€ê³ ëŒ€ê°�í™” 불가능한 í–‰ë ¬ì�˜ 예를 들어ë�¼. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$082e17d6-a51b-4c2e-b61c-116465962f53„§cell_idÙ$082e17d6-a51b-4c2e-b61c-116465962f53¤code¤true¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$ab514fd2-62ef-4664-9454-f052895c4498„§cell_idÙ$ab514fd2-62ef-4664-9454-f052895c4498¤codeÙOmd""" (4) 아래ì�˜ 매트ë¦ìŠ¤ê°€ positive definite ì�¸ì§€ íŒ�단하ë�¼. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$4b423561-ca96-4a7b-a515-41072c624d49„§cell_idÙ$4b423561-ca96-4a7b-a515-41072c624d49¤code¤true¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$1166210b-1735-4e1c-87aa-a461ec81dcbf„§cell_idÙ$1166210b-1735-4e1c-87aa-a461ec81dcbf¤codeÙNmd""" $\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 3 & 2 & 5 \end{bmatrix}$ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$ab312f30-9096-484d-b7bf-f95f80ea8201„§cell_idÙ$ab312f30-9096-484d-b7bf-f95f80ea8201¤codeÙ"let A = [2 2; 2 1] eigen(A) end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$8e3b2f81-3832-468a-935c-687c755b1f67„§cell_idÙ$8e3b2f81-3832-468a-935c-687c755b1f67¤codeÙ/md""" (í’€ì�´) ìƒ�ëžµ / ê°•ì�˜ë…¸íЏ ì°¸ê³ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$02cb7333-0f30-49c9-babb-ae142d9efe2b„§cell_idÙ$02cb7333-0f30-49c9-babb-ae142d9efe2b¤code¤true¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$145878ce-ab24-437d-bc37-fecb342553fe„§cell_idÙ$145878ce-ab24-437d-bc37-fecb342553fe¤code²md""" (í’€ì�´) """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$2efcd746-0ece-425b-9204-464bb35182ee„§cell_idÙ$2efcd746-0ece-425b-9204-464bb35182ee¤codeÙ_md""" 아래ì�˜ 코드를 ì�´ìš©í•˜ì—¬ ì�´ë¯¸ì§€ìž�료를 ì�½ì–´ì˜¤ê³ 물ì�Œì—� 답하ë�¼. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$e17e2ff1-0b58-4b34-8747-1785aa104a8b„§cell_idÙ$e17e2ff1-0b58-4b34-8747-1785aa104a8b¤codeÙmd""" (3) SVD를 ì�´ìš©í•˜ì—¬ 매트ë¦ìФ $\begin{bmatrix}2 & 2 & 3\\ 2 & 1 & -5 \\ 3 & -5 & 3 \end{bmatrix}$ ì�˜ ê³ ìœ ë²¡í„°í–‰ë ¬ê³¼ ê³ ìœ ê°’ì�„ 구하여ë�¼. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$714f6a24-36b4-471e-a90e-09ac1ef83ef8„§cell_idÙ$714f6a24-36b4-471e-a90e-09ac1ef83ef8¤codeÙ$false # ê°•ì�˜ë…¸íЏì—� 반례있ì�Œ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$db16c741-2987-4a4d-9723-beb7c07f9d29„§cell_idÙ$db16c741-2987-4a4d-9723-beb7c07f9d29¤code¤true¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$5b2da4a9-c8ae-49cf-a763-a8d91adf91fe„§cell_idÙ$5b2da4a9-c8ae-49cf-a763-a8d91adf91fe¤codeÙtmd""" (보충학습1) ìƒ�삼ê°�í–‰ë ¬, 하삼ê°�í–‰ë ¬ì�€ 대ê°�ì„ ì�˜ ì›�소가 그대로 ê³ ìœ ê°’ì�´ ë�œë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$79d2562f-94ff-42d7-a2e7-491dfdb7bcfc„§cell_idÙ$79d2562f-94ff-42d7-a2e7-491dfdb7bcfc¤codeÙ\md""" (1) $n\times n$ í–‰ë ¬ì�€ í•ìƒ� 서로 다른 $n$ê°œì�˜ ê³ ìœ ê°’ì�„ 가진다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$b67cf197-da63-427b-a5df-ebc1fcb5124e„§cell_idÙ$b67cf197-da63-427b-a5df-ebc1fcb5124e¤codeÙ9md""" $\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$863513de-f616-4a24-bade-6a2929019777„§cell_idÙ$863513de-f616-4a24-bade-6a2929019777¤codeÙ›md""" (지수) 만약ì—� ${\bf Z}={\bf U}{\bf D}$로 ì •ì�˜í•œë‹¤ë©´ ${\bf Z}$를 ì�´ìš©í•˜ì—¬ ì–¸ì œë‚˜ ${\bf X}$를 reconstruction í• ìˆ˜ 있다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$8b415272-f97f-4e6b-838e-417043000eaf„§cell_idÙ$8b415272-f97f-4e6b-838e-417043000eaf¤codeÙfmd""" (3) svd를 ì�´ìš©í•˜ì—¬ ì›�본ì�´ë¯¸ì§€ì�˜ 10%ë§Œ ë�°ì�´í„°ë¥¼ 사용하게ë�” 압축하ë�¼. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$cfcd2836-0179-4d90-a757-4aa4b8580622„§cell_idÙ$cfcd2836-0179-4d90-a757-4aa4b8580622¤codeÙOmd""" (ë�™í•˜) ${\bf X}^\top{\bf X}$ì�˜ ê³ ìœ ê°’ë¶„í•´ëŠ” 존재한다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$9971e318-71ec-4102-856a-97789436e85a„§cell_idÙ$9971e318-71ec-4102-856a-97789436e85a¤codeÙ†md""" (ì„ ì² ) 만약ì—� ${\bf Z}={\bf U}{\bf D}$로 ì •ì�˜í•œë‹¤ë©´ ${\bf X}^\top{\bf X}={\bf Z}^\top {\bf Z}$ ê°€ 성립한다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$a7b42a7b-a5dd-4d96-b863-548aa151fdbd„§cell_idÙ$a7b42a7b-a5dd-4d96-b863-548aa151fdbd¤code·false # 서로다른 X ¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$bf8015b5-58b6-4451-8080-836918494854„§cell_idÙ$bf8015b5-58b6-4451-8080-836918494854¤codeÙwmd""" - ì�´ ë¬¸ì œì—�서 ${\bf X}$는 ì•„ì�´ë¦¬ìФìž�료ì�˜ì—�서 ì •ì�˜í•œ (150,4) matrix로 í•œì •í•˜ì—¬ 푼다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$d2e49602-7d16-4bca-86af-4d4ddde67e6a„§cell_idÙ$d2e49602-7d16-4bca-86af-4d4ddde67e6a¤codeÙ™md""" (7) ëª¨ë“ ì›�소가 실수ì�¸ ìž„ì�˜ì�˜ í–‰ë ¬ ${\bf X}$ì—� 대하여 ${\bf X}^\top{\bf X}$는 ì�˜ ëª¨ë“ ê³ ìœ ê°’ì�€ 0 ë˜�는 양수ì�´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$ce888ed5-46a4-4064-9579-9ad04bb7dc6d„§cell_idÙ$ce888ed5-46a4-4064-9579-9ad04bb7dc6d¤codeÙZlet A = [1 22 33 4 -5 6 0 8 9 ] λ=eigvals(A) sum(λ),tr(A),prod(λ),det(A) end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$ae8dfccd-456f-4e5e-b9dd-c8cbf75270fc„§cell_idÙ$ae8dfccd-456f-4e5e-b9dd-c8cbf75270fc¤codeÙxmd""" (2) ìž„ì�˜ì�˜ 대ì¹í–‰ë ¬ (허수ì�˜ ì›�소를 가질 수 있ì�Œ)ì�€ í•ìƒ� 실수ì�¸ ê³ ìœ ê°’ì�„ 가진다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$b479109c-28d6-4d5b-9435-3711dc16d949„§cell_idÙ$b479109c-28d6-4d5b-9435-3711dc16d949¤codeÙ/md""" (í’€ì�´) ìƒ�ëžµ / ê°•ì�˜ë…¸íЏ ì°¸ê³ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$544504e9-03a3-4be0-8cab-33fb4ccf2132„§cell_idÙ$544504e9-03a3-4be0-8cab-33fb4ccf2132¤codeÙfmd""" (í’€ì�´) 실대ì¹í–‰ë ¬ì�´ ì�Œì�˜ ê³ ìœ ê°’ì�„ 가지므로 positive definiteì�´ 아니다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$a3753790-56b3-4a1b-ba09-91a7c31a4238„§cell_idÙ$a3753790-56b3-4a1b-ba09-91a7c31a4238¤codeÙPmd""" (ë¯¼ì •) ${\bf X}^\top{\bf X}$ì�˜ ëª¨ë“ ê³ ìœ ê°’ì�€ 실수ì�´ë‹¤. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$f21c5621-e720-4250-a914-0b99f297f0f4„§cell_idÙ$f21c5621-e720-4250-a914-0b99f297f0f4¤code®Plots.plotly()¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$2085f751-0248-47ae-984c-a7cb226ab5f8„§cell_idÙ$2085f751-0248-47ae-984c-a7cb226ab5f8¤codeÙ@md""" $\begin{bmatrix}1 & 2 & 3\\ 1 & 2 & 3 \end{bmatrix}$ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$e34e42cf-7f1e-40be-a8f8-aaec153910ab„§cell_idÙ$e34e42cf-7f1e-40be-a8f8-aaec153910ab¤codeÙXmd""" (í’€ì�´) ê³ ìœ ê°’ì�˜ í•©ê³¼ 대ê°�ì„ ì›�소ì�˜ í•©ì�´ 모ë‘� 1+3+5로 같다. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$c42ebadd-726a-4746-80f1-d18d9b7fb672„§cell_idÙ$c42ebadd-726a-4746-80f1-d18d9b7fb672¤codeÙ€md""" (2) 벡터 ${\bf v}=[3,4]^\top$ 는 매트ë¦ìФ $\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ì�˜ ê³ ìœ ë²¡í„°ì�¸ê°€? """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$630d5597-77cd-4fcf-90a5-c3f814eee5fe„§cell_idÙ$630d5597-77cd-4fcf-90a5-c3f814eee5fe¤codeÙ#let A = [1 2 ; 2 3] eigen(A) end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$1609eb31-c8f0-40c2-a584-768a453f6720„§cell_idÙ$1609eb31-c8f0-40c2-a584-768a453f6720¤codeÙÝmd""" (3) ì–´ë– í•œ í–‰ë ¬ ${\bf A}$ê°€ 실대ì¹í–‰ë ¬ì�´ê³ ëª¨ë“ ê³ ìœ ê°’ì�´ 양수ë�¼ë©´ 0벡터가 아닌 ìž„ì�˜ì�˜ 벡터 $\bf y$ì—� 대하여 아래ì‹�ì�„ í•ìƒ� 만족한다. ${\bf y}^\top{\bf A}{\bf y}> 0$ """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$596eaef1-a31c-4d7c-8805-7151deb6759b„§cell_idÙ$596eaef1-a31c-4d7c-8805-7151deb6759b¤codeÙmd""" (10) 아래 í–‰ë ¬ì�˜ ëª¨ë“ ê³ ìœ ê°’ì�˜ ê³±ì�„ 구하ë�¼. í–‰ë ¬ì�˜ ëª¨ë“ ëŒ€ê°�ì„ ì›�소ì�˜ 곱과 비êµ�하ë�¼. """¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$3ebbea1e-5640-4be6-afa8-cb1e57c28814„§cell_idÙ$3ebbea1e-5640-4be6-afa8-cb1e57c28814¤codeÚzhtml"""
(슬기) ${\bf X}$는 $150\times 4$ matrix �므로 ${\bf X}$� rank는 기�해야 (=아무리 커�) 4�다.
(í’€ì�´) ê³ ìœ ê°’ì�˜ 곱과 대ê°�ì„ ì›�소ì�˜ ê³±ì�´ 모ë‘� 1×3×5로 같다.
(8) 중복ë�œ ê³ ìœ ê°’ì�„ ê°€ì§€ê³ ëŒ€ê°�í™” 불가능한 í–‰ë ¬ì�˜ 예를 들어ë�¼.
(4) 아래ì�˜ 매트ë¦ìŠ¤ê°€ positive definite ì�¸ì§€ íŒ�단하ë�¼.
$$\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 3 & 2 & 5 \end{bmatrix}$$
(í’€ì�´) ìƒ�ëžµ / ê°•ì�˜ë…¸íЏ ì°¸ê³
(풀�)
아래ì�˜ 코드를 ì�´ìš©í•˜ì—¬ ì�´ë¯¸ì§€ìž�료를 ì�½ì–´ì˜¤ê³ 물ì�Œì—� 답하ë�¼.
(3) SVD를 ì�´ìš©í•˜ì—¬ 매트ë¦ìФ
$$\begin{bmatrix}2 & 2 & 3\\ 2 & 1 & -5 \\ 3 & -5 & 3 \end{bmatrix}$$
ì�˜ ê³ ìœ ë²¡í„°í–‰ë ¬ê³¼ ê³ ìœ ê°’ì�„ 구하여ë�¼.
(보충학습1) ìƒ�삼ê°�í–‰ë ¬, 하삼ê°�í–‰ë ¬ì�€ 대ê°�ì„ ì�˜ ì›�소가 그대로 ê³ ìœ ê°’ì�´ ë�œë‹¤.
(1) $n\times n$ í–‰ë ¬ì�€ í•ìƒ� 서로 다른 $n$ê°œì�˜ ê³ ìœ ê°’ì�„ 가진다.
$$\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$$
(지수) 만약ì—� ${\bf Z}={\bf U}{\bf D}$로 ì •ì�˜í•œë‹¤ë©´ ${\bf Z}$를 ì�´ìš©í•˜ì—¬ ì–¸ì œë‚˜ ${\bf X}$를 reconstruction í• ìˆ˜ 있다.
(3) svd를 �용하여 �본�미지� 10%만 ��터를 사용하게� 압축하�.
(ë�™í•˜) ${\bf X}^\top{\bf X}$ì�˜ ê³ ìœ ê°’ë¶„í•´ëŠ” 존재한다.
(ì„ ì² ) 만약ì—� ${\bf Z}={\bf U}{\bf D}$로 ì •ì�˜í•œë‹¤ë©´ ${\bf X}^\top{\bf X}={\bf Z}^\top {\bf Z}$ ê°€ 성립한다.
ì�´ ë¬¸ì œì—�서 ${\bf X}$는 ì•„ì�´ë¦¬ìФìž�료ì�˜ì—�서 ì •ì�˜í•œ (150,4) matrix로 í•œì •í•˜ì—¬ 푼다.
(7) ëª¨ë“ ì›�소가 실수ì�¸ ìž„ì�˜ì�˜ í–‰ë ¬ ${\bf X}$ì—� 대하여 ${\bf X}^\top{\bf X}$는 ì�˜ ëª¨ë“ ê³ ìœ ê°’ì�€ 0 ë˜�는 양수ì�´ë‹¤.
(2) ìž„ì�˜ì�˜ 대ì¹í–‰ë ¬ (허수ì�˜ ì›�소를 가질 수 있ì�Œ)ì�€ í•ìƒ� 실수ì�¸ ê³ ìœ ê°’ì�„ 가진다.
(í’€ì�´) ìƒ�ëžµ / ê°•ì�˜ë…¸íЏ ì°¸ê³
(í’€ì�´) 실대ì¹í–‰ë ¬ì�´ ì�Œì�˜ ê³ ìœ ê°’ì�„ 가지므로 positive definiteì�´ 아니다.
(ë¯¼ì •) ${\bf X}^\top{\bf X}$ì�˜ ëª¨ë“ ê³ ìœ ê°’ì�€ 실수ì�´ë‹¤.
$$\begin{bmatrix}1 & 2 & 3\\ 1 & 2 & 3 \end{bmatrix}$$
(í’€ì�´) ê³ ìœ ê°’ì�˜ í•©ê³¼ 대ê°�ì„ ì›�소ì�˜ í•©ì�´ 모ë‘� 1+3+5로 같다.
(2) 벡터 ${\bf v}=[3,4]^\top$ 는 매트ë¦ìФ $\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ì�˜ ê³ ìœ ë²¡í„°ì�¸ê°€?
(3) ì–´ë– í•œ í–‰ë ¬ ${\bf A}$ê°€ 실대ì¹í–‰ë ¬ì�´ê³ ëª¨ë“ ê³ ìœ ê°’ì�´ 양수ë�¼ë©´ 0벡터가 아닌 ìž„ì�˜ì�˜ 벡터 $\bf y$ì—� 대하여 아래ì‹�ì�„ í•ìƒ� 만족한다.
$${\bf y}^\top{\bf A}{\bf y}> 0$$
(10) 아래 í–‰ë ¬ì�˜ ëª¨ë“ ê³ ìœ ê°’ì�˜ ê³±ì�„ 구하ë�¼. í–‰ë ¬ì�˜ ëª¨ë“ ëŒ€ê°�ì„ ì›�소ì�˜ 곱과 비êµ�하ë�¼.
(6) 대ê°�í™” 불가능한 í–‰ë ¬ì�˜ 예를 들어ë�¼. (ë¬¸ì œ8ê³¼ ê²¹ì³�서 ì‚ì œí•©ë‹ˆë‹¤)
(1) 아래 매트ë¦ìФì�˜ ê³ ìœ ê°’ì�„ 구해ë�¼.
$$\begin{bmatrix}2 & 2 \\ 2 & 1 \end{bmatrix}$$
(8) 실대ì¹í–‰ë ¬ ${\bf A}$ì�˜ ê³ ìœ ê°’ì�´ $\lambda_1,\lambda_2,\dots,\lambda_n$ ë�¼ë©´ ${\bf A}^2$ì�˜ ê³ ìœ ê°’ì�€ $\lambda_1^2, \lambda_2^2,\dots,\lambda_n^2$ ì�´ë‹¤.
(풀�)
(2) (1)� 결과를 �용하여 위� �미지를 sharpening 하�.
(성재) ${\bf X}=\sum_{i=1}^{4}U_i d_i V_i^\top$ 로 표현가능한 벡터 $U_i,V_i$와 스칼ë�¼ $d_i$ê°€ 존재한다. (ì�´ë•Œ $\{U_1,\dots, U_4\}$는 서로 ì§�êµ�í•˜ê³ $\{V_1,\dots,V_4\}$ ì—시 서로 ì§�êµ�한다.)
(í’€ì�´) ê°•ì�˜ë…¸íЏ ì°¸ê³
(7) 중복ë�œ ê³ ìœ ê°’ì�„ 가지지만 대ê°�í™” 가능한 í–‰ë ¬ì�˜ 예를 들어ë�¼.
아래를 ì�´ìš©í•˜ì—¬ ì•„ì�´ë¦¬ìФ ìž�료를 loadí•˜ê³ ë¬¼ì�Œì—� 답하ë�¼.
(2) 다ì�Œ 중 PCAì—� 대하여 올바른 ë¶„ì„�ì�„ 한 사람ì�„ 모ë‘� 골ë�¼ë�¼. (모ë‘� ë§žì¶œ 경우만 ì •ë‹µìœ¼ë¡œ ì�¸ì •)
(í’€ì�´) ê°•ì�˜ë…¸íЏ ì°¸ê³
(9) 실대ì¹í–‰ë ¬ ${\bf A}$ì�˜ ê³ ìœ ë²¡í„°ëŠ” ${\bf A}^2$ì�˜ ê³ ìœ ë²¡í„°ê°€ ë�œë‹¤.
(풀�)
(9) 아래 í–‰ë ¬ì�˜ ëª¨ë“ ê³ ìœ ê°’ì�˜ í•©ì�„ 구하ë�¼. í–‰ë ¬ì�˜ ëª¨ë“ ëŒ€ê°�ì„ ì›�소ì�˜ í•©ê³¼ 비êµ�하ë�¼.
note: $tr({\bf X}^\top{\bf X}) = tr({\bf Z}'{\bf Z})$� 성립함.
(4) 0벡터가 아닌 ìž„ì�˜ì�˜ 벡터 ${\bf y}$ì—� 대하여 아래ì‹�ì�„ 만족하는 í–‰ë ¬ ${\bf A}$는 í•ìƒ� ì–‘ì�˜ ê³ ìœ ê°’ì�„ 가진다.
$${\bf y}^\top{\bf A}{\bf y}> 0$$
(보충학습2) trace, determinant, ê³ ìœ ê°’ê³¼ ê´€ë ¨ë�œ 성질