Principal Component Analysis; PCA

はじめての『はじめてのパターン認識』第9章 This is a general introduction of Principal Component Analysis; PCA for machine learning, which includes vector space and subspace, PCA and singular value decomposition; SVD.

Transcript

1. ෦෼ۭؒͱओ੒෼෼ੳ ͸͡Ίͯͷύλʔϯೝࣝྠಡձ ޻౻঑  ୈষ
   ෦෼ۭؒ๏
2. 065-*/& ·ͱΊ ओ੒෼෼ੳ ಛҟ஋෼ղ ෦෼ۭؒ ෦෼ۭؒ๏
3. 065-*/& ·ͱΊ ओ੒෼෼ੳ ಛҟ஋෼ղ ෦෼ۭؒ ෦෼ۭؒ๏ ࿩͠·ͤΜ
4. 065-*/& ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ ਖ਼ن௚ަجఈ ෦෼ۭؒ ຕ ຕ ຕ ·ͱΊ ຕ
5. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
   ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜ΕΛҰͭҰͭઆ໌͍ͯ͘͠ʜ
6. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
   ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} V ϕΫτϧۭؒɹ͸ͦΕͧΕ௚ߦ͢Δɹݸͷ͔࣠ΒͰ͖ͯΔ d Vd=3 ݸ ݸ ݸ 1 2 3
7. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
   ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} Ұ࣍ಠཱͳϕΫτϧͱ͸ɼྫ͑͹ʜ Vd=3 Vd=3
   ্࣠ͷϕΫτϧ
   ্࣠ͷϕΫτϧͱ
   ɹɹฏ໘͔Β֎ΕͨϕΫτϧ x1 x3 x2 x1 x3 x2 x1 x2
8. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
   ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} Ұ࣍ಠཱͳϕΫτϧͱ͸ɼྫ͑͹ʜ Vd=3 Vd=3 εΧϥʔഒͱϕΫτϧͷ଍͠ࢉͰ
   දݱͰ͖ͳ͍Α͏ͳϕΫτϧͷ૊ x1 ⟹ a1 x1 x1 ⟹ x1 + x2 εΧϥʔഒ ଍͠ࢉ x1 x3 x2 x1 x3 x2
9. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
   ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} Ұ࣍ಠཱͳϕΫτϧͱ͸ɼྫ͑͹ʜ Vd=3 Vd=3 εΧϥʔഒͱϕΫτϧͷ଍͠ࢉͰ
   දݱͰ͖ͳ͍Α͏ͳϕΫτϧͷ૊ ɹ࣍ݩͷϕΫτϧۭؒͰ͸ɹݸͷ
   Ұ࣍ಠཱͳϕΫτϧ͕ଘࡏ͢Δɽ d d x1 x3 x2 x1 x3 x2
10. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} Ұ࣍ಠཱͳϕΫτϧ εΧϥʔഒͱϕΫτϧͷ଍͠ࢉͰ
    දݱͰ͖ͳ͍Α͏ͳϕΫτϧͷ૊ εΧϥʔഒͱϕΫτϧͷ଍͠ࢉ
    ͰදݱͰ͖ΔϕΫτϧ Vd=3 Vd=3 ɹɹฏ໘্ͷ
    ϕΫτϧ
    
    ฏ໘্ͷ
    ϕΫτϧ Ұ࣍ैଐ ઢܗैଐ ͳϕΫτϧ x1 x3 x2 x1 x3 x2 x1 x2 x2 x3
11. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} Ұ࣍ಠཱͳϕΫτϧ εΧϥʔഒͱϕΫτϧͷ଍͠ࢉͰ
    දݱͰ͖ͳ͍Α͏ͳϕΫτϧͷ૊ εΧϥʔഒͱϕΫτϧͷ଍͠ࢉ
    ͰදݱͰ͖ΔϕΫτϧ Vd=3 Vd=3 εΧϥʔഒ Ұ࣍ैଐ ઢܗैଐ ͳϕΫτϧ x1 x3 x2 x1 x3 x2 ɹɹฏ໘্ͷ
    ϕΫτϧ x1 x2
    
    ฏ໘্ͷ
    ϕΫτϧ x2 x3
12. x2 x1 x3 ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} Ұ࣍ಠཱͳϕΫτϧ εΧϥʔഒͱϕΫτϧͷ଍͠ࢉͰ
    දݱͰ͖ͳ͍Α͏ͳϕΫτϧͷ૊ εΧϥʔഒͱϕΫτϧͷ଍͠ࢉ
    ͰදݱͰ͖ΔϕΫτϧ Ұ࣍ैଐ ઢܗैଐ ͳϕΫτϧ Vd=3 Vd=3 ϕΫτϧͷ଍͠ࢉ x2 x1 x3 ɹɹฏ໘্ͷ
    ϕΫτϧ x1 x2
    
    ฏ໘্ͷ
    ϕΫτϧ x2 x3
13. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} εΧϥʔഒͱϕΫτϧͷ଍͠ࢉ
    ͰදݱͰ͖ΔϕΫτϧ Ұ࣍ैଐ ઢܗैଐ ͳϕΫτϧΛఆࣜԽ͢Δͱʜ Vd=3 x2 x1 x3 ɹɹฏ໘্ͷ
    ϕΫτϧ x1 x2 ۭؒɹɹʹؚ·ΕΔ೚ҙͷϕΫτϧɹ͸ɼ
    ۭؒɹɹதʹଘࡏ͢ΔҰ࣍ಠཱͳϕΫτϧ
    ɹɹɹɹʢجఈʣͷઢܕ݁߹ͰදݱͰ͖Δɽ Vd=3 v x1 , x2 , x3 Vd=3 ೚ҙఆ਺ جఈ v = a1 x1 + a2 x2 + a3 x3 = 3 ∑ i=1 ai xi ai xi xi ai
14. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} εΧϥʔഒͱϕΫτϧͷ଍͠ࢉ
    ͰදݱͰ͖ΔϕΫτϧ Ұ࣍ैଐ ઢܗैଐ ͳϕΫτϧΛఆࣜԽ͢Δͱʜ Vd=3 x2 x1 x3 ɹɹฏ໘্ͷ
    ϕΫτϧ x1 x2 ೚ҙఆ਺ v = d ∑ i=1 ai xi ai xi xi ai ࣍ݩͷϕΫτϧۭؒɹͰҰൠԽ͢Δͱʜ d V جఈ
15. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜͜·Ͱཧղ্ͨ͠Ͱ෦෼ۭؒΛಋೖ͠·͢ʜ ϕΫτϧۭ͕ؒ೚ҙͷ࣍ݩͰ͋Δ৔߹Λߟ͑Δલʹ·ͣ͸ɹɹɹͰʜ Vd=3 Vd=3 x1 x3 x2 W Λɹͱɹ͔Β੒ΔϕΫτϧશମͱͨ࣌͠ɼ
    ͸ϕΫτϧۭؒɹɹʹแؚ͞ΕΔɽ W x1 x2 Vd=3 ͜ͷ࣌ɼ࣍ͷ৚݅Λຬ͔ͨ͢Λߟ͑Δɽ Vd=3 Wd=2
16. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜͜·Ͱཧղ্ͨ͠Ͱ෦෼ۭؒΛಋೖ͠·͢ʜ ϕΫτϧۭ͕ؒ೚ҙͷ࣍ݩͰ͋Δ৔߹Λߟ͑Δલʹ·ͣ͸ɹɹɹͰʜ Vd=3 Vd=3 x1 x3 x2 ͜ͷ࣌ɼ࣍ͷ৚݅Λຬ͔ͨ͢Ͳ͏͔Λߟ͑Δɽ (1) W ≠ ∅ (2) x1 , x2 ∈ W ⟹ x1 + x2 ∈ W (3) x1 ∈ W, λ ∈ R ⟹ λx1 ∈ W Wd=2
17. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜͜·Ͱཧղ্ͨ͠Ͱ෦෼ۭؒΛಋೖ͠·͢ʜ ϕΫτϧۭ͕ؒ೚ҙͷ࣍ݩͰ͋Δ৔߹Λߟ͑Δલʹ·ͣ͸ɹɹɹͰʜ Vd=3 Vd=3 x1 x3 x2 ͜ͷ࣌ɼ࣍ͷ৚݅Λຬ͔ͨ͢Ͳ͏͔Λߟ͑Δɽ (1) W ≠ ∅ ɹ্࣠ͱɹ্࣠ͷೋछྨͷϕΫτϧʹΑΓ
    දݱ͞ΕΔϕΫτϧ͕ଘࡏ͢ΔͷͰɼɹ͸
    ۭू߹Ͱ͸ͳ͍ɽ x1 x2 W Wd=2
18. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜͜·Ͱཧղ্ͨ͠Ͱ෦෼ۭؒΛಋೖ͠·͢ʜ ϕΫτϧۭ͕ؒ೚ҙͷ࣍ݩͰ͋Δ৔߹Λߟ͑Δલʹ·ͣ͸ɹɹɹͰʜ Vd=3 ͜ͷ࣌ɼ࣍ͷ৚݅Λຬ͔ͨ͢Ͳ͏͔Λߟ͑Δɽ (2) x1 , x2 ∈ W ⟹ x1 + x2 ∈ W Vd=3 x1 x3 x2 ɹ্࣠ͱɹ্࣠ͷೋछྨͷϕΫτϧͰදݱ
    ͞ΕΔɹɹฏ໘্ͷશ͕ͯɹʹؚ·ΕΔͷͰ
    ਖ਼͍͠ɽ x1 x2 x1 x2 W Wd=2
19. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜͜·Ͱཧղ্ͨ͠Ͱ෦෼ۭؒΛಋೖ͠·͢ʜ ϕΫτϧۭ͕ؒ೚ҙͷ࣍ݩͰ͋Δ৔߹Λߟ͑Δલʹ·ͣ͸ɹɹɹͰʜ Vd=3 Vd=3 x1 x3 x2 ͜ͷ࣌ɼ࣍ͷ৚݅Λຬ͔ͨ͢Ͳ͏͔Λߟ͑Δɽ (3) x1 ∈ W, λ ∈ R ⟹ λx1 ∈ W ɹ্࣠ͱɹ্࣠ͷೋछྨͷϕΫτϧͦΕͧΕͷ
    εΧϥʔഒ΋·ͨɹ্ͷϕΫτϧͳͷͰਖ਼͍͠ɽ x1 x2 W Wd=2
20. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜͜·Ͱཧղ্ͨ͠Ͱ෦෼ۭؒΛಋೖ͠·͢ʜ ϕΫτϧۭ͕ؒ೚ҙͷ࣍ݩͰ͋Δ৔߹Λߟ͑Δલʹ·ͣ͸ɹɹɹͰʜ Vd=3 Vd=3 x1 x3 x2 ͜ͷ࣌ɼ࣍ͷ৚݅Λຬ͔ͨ͢Ͳ͏͔Λߟ͑Δɽ (2) x1 , x2 ∈ W ⟹ x1 + x2 ∈ W (3) x1 ∈ W, λ ∈ R ⟹ λx1 ∈ W ্ه৚݅Λ߹ΘͤΔͱϕΫτϧۭؒɹͷҰ࣍ಠཱͳ
    ϕΫτϧͷઢܕ݁߹΋ɹۭؒͷ੒෼Ͱ͋Δͱݴ͑Δ W W Wd=2
21. ෦෼ۭؒ ෦෼ۭؒ ɹ࣍ݩϕΫτϧۭؒɹͷ෦෼ۭؒɹ͸ɼɹʹؚ·ΕΔҰ࣍ಠཱͳϕΫτϧɼ
    ɹɹɹɹɹɹɹɹɹͱ೚ҙͷఆ਺ɹɹɹɹɹɹΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͢Δɽ d V W x1 , x2 , . . . , xr (r ≤ d) V a1 , a2 , . . . , ad W = {a1 x1 + a2 x2 + . . . + ar xr |ai ∈ R, i = 1,2,...,r} ͜͜·Ͱཧղ্ͨ͠Ͱ෦෼ۭؒΛಋೖ͠·͢ʜ Vd=3 x1 x3 x2 ཁૉ͕θϩͰͳ͍ۭؒɹ͕ɹ࣍ݩ
    ϕΫτϧۭؒɹͷ෦෼ۭؒͰ͋Δ
    ͱ͍͏ͷ͸ɹΑΓ΋খ͍͞ɹݸͷ
    جఈͷઢܕ݁߹Ͱ೚ҙͷϕΫτϧ
    ΛҰҙʹදݱͰ͖ΔϕΫτϧۭؒ
    Ͱ͋Δͱ͍͏͜ͱɽ W d V d r Wd=2
22. ෦෼ۭؒ ਖ਼ن௚ަجఈ ਖ਼ن௚ަجఈͱ͸௕͕͞ɹͰ ਖ਼نԽ͞Ε͍ͯΔ ޓ͍ʹ௚ަ͍ͯ͠Δ ಺ੵ͕ɹ
    Α͏ͳجఈ ઢܗ݁߹ͰશͯΛදͤΔΑ͏ͳඞཁ࠷௿ݶͷϕΫτϧͷू߹ Ͱ͋Δɽ 1 0 1 Vd=3 x1 x3 x2 Ұ࣍ಠཱͰ͋ΔجఈɹɹɹɹΛ௕͕͞ɹͰ͔ͭ
    ௚ަ͢ΔΑ͏ʹઃఆͨ͠΋ͷ͕ਖ਼ن௚ަجఈɽ x1 , x2 , x3 1 ∥x1 ∥ = ∥x2 ∥ = ∥x3 ∥ = 1 xT 1 x2 = xT 2 x3 = xT 3 x1 = 0 {v ∈ Vd=3 |v = 3 ∑ i=1 ai xi }
23. ෦෼ۭؒ ਖ਼ن௚ަجఈ ਖ਼ن௚ަجఈͱ͸௕͕͞ɹͰ ਖ਼نԽ͞Ε͍ͯΔ ޓ͍ʹ௚ަ͍ͯ͠Δ ಺ੵ͕ɹ
    Α͏ͳجఈ ઢܗ݁߹ͰશͯΛදͤΔΑ͏ͳඞཁ࠷௿ݶͷϕΫτϧͷू߹ Ͱ͋Δɽ 1 0 1 Vd=3 x1 x3 x2 ࠨਤͷΑ͏ͳϕΫτϧҎ֎ʹ΋ࡾ࣍ݩͷ
    ϕΫτϧۭؒʹ͓͍ͯࡾͭͷϕΫτϧ͕
    ༩͑ΒΕͨ࣌ʹ͸ͦΕΛݩʹࡾͭͷجఈ
    Λ࡞Δ͜ͱ͕Ͱ͖Δɽ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ
24. ෦෼ۭؒ Vd=3 x1 x3 x2 άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ  ҰͭͷϕΫτϧΛબΜͰਖ਼نԽ͢Δ ௕͞Λʹ͢Δ ɽ n1 = x1 ∥x1 ∥ = x1 12 + 02 + 12 = 1 2 ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) ྫ x1 = ( 1 0 1 )
25. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ  ҰͭͷϕΫτϧΛબΜͰਖ਼نԽ͢Δ ௕͞Λʹ͢Δ ɽ n1 = x1 ∥x1 ∥ = x1 12 + 02 + 12 = 1 2 ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) ྫ n1 = 1 2 ( 1 0 1 ) x1 = ( 1 0 1 ) Vd=3 n1 x3 x2
26. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ  ɹͷ৘ใΛ࢖ͬͯೋͭ໨ͷجఈΛ࡞Δɽ ྫ n1 = 1 2 ( 1 0 1 ) n1 x1 = ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) ˜ n2 = x2 − 1 ∑ j=1 (nT j x2) nj = (x2 − (nT 1 x2) n1) = ( 2 1 0 ) − 1 2 2 (1 0 1) ( 2 1 0 ) ( 1 0 1 ) = ( 1 1 −1 ) Vd=3 n1 x3 x2
27. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ  ɹͷ৘ใΛ࢖ͬͯೋͭ໨ͷجఈΛ࡞Δɽ ྫ n1 = 1 2 ( 1 0 1 ) n1 x1 = ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) n2 = 1 3 ( 1 1 −1 ) ˜ n2 = x2 − 1 ∑ j=1 (nT j x2) nj = (x2 − (nT 1 x2) n1) = ( 2 1 0 ) − 1 2 2 (1 0 1) ( 2 1 0 ) ( 1 0 1 ) = ( 1 1 −1 ) Vd=3 n1 x3 n2
28. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ  ɹɹͷ৘ใΛ࢖ͬͯࡾͭ໨ͷجఈΛ࡞Δɽ ˜ n3 = x3 − 2 ∑ j=1 (nT j x3) nj = (x3 − (nT 1 x3) n1 − (nT 2 x3) n2) ྫ n1 , n2 x1 = ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) n1 = 1 2 ( 1 0 1 ) n2 = 1 3 ( 1 1 −1 ) = ( 0 1 1 ) − 1 2 2 (1 0 1) ( 0 1 1 ) ( 1 0 1 ) − 1 3 2 (1 1 − 1) ( 0 1 1 ) ( 1 1 −1 ) = ( −1 2 1 ) Vd=3 n1 x3 n2
29. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ  ɹɹͷ৘ใΛ࢖ͬͯࡾͭ໨ͷجఈΛ࡞Δɽ ˜ n3 = x3 − 2 ∑ j=1 (nT j x3) nj = (x3 − (nT 1 x3) n1 − (nT 2 x3) n2) ྫ n1 , n2 x1 = ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) n1 = 1 2 ( 1 0 1 ) n2 = 1 3 ( 1 1 −1 ) = ( 0 1 1 ) − 1 2 2 (1 0 1) ( 0 1 1 ) ( 1 0 1 ) − 1 3 2 (1 1 − 1) ( 0 1 1 ) ( 1 1 −1 ) = ( −1 2 1 ) n3 = 1 2 3 ( −1 2 1 ) Vd=3 n1 n3 n2
30. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ ௚ަ͍ͯ͠Δ͔֬ೝͯ͠ΈΔʜ ྫ x1 = ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) n1 = 1 2 ( 1 0 1 ) n2 = 1 3 ( 1 1 −1 ) n3 = 1 2 3 ( −1 2 1 ) nT 1 n2 = 1 6 − 1 6 = 0 nT 2 n3 = − 1 6 + 2 6 − 1 6 = 0 nT 3 n1 = − 1 2 6 + 1 2 6 = 0 Vd=3 n1 n3 n2
31. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ ͪͳΈʹ΋͏ҰͭͷϕΫτϧΛ͖࣋ͬͯͯಉ༷ͷܭࢉΛ͢Δͱʜ ྫ x1 = ( 1 0 1 ) x2 = ( 2 1 0 ) x3 = ( 0 1 1 ) n1 = 1 2 ( 1 0 1 ) n2 = 1 3 ( 1 1 −1 ) n3 = 1 2 3 ( −1 2 1 ) x4 = ( 1 0 0 ) ⟹ ˜ n4 = 1 12 ( 1 −2 −1 ) ͜Ε͸໌Β͔ʹɹͷఆ਺ഒͰ͋ΔͨΊ৽͍͠جఈʹ͸ͳΒͳ͍ n3 Vd=3 n1 n3 n2
32. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ ೚ҙͷɹ࣍ݩͰಉ༷ʹ੒ཱ͢Δ͔Λ֬ೝ͢ΔͨΊʹ਺ֶతؼೲ๏ΛؤுΔʜ k (1) i = 1 ͷ࣌ʜ n1 = x1 ∥x1 ∥ (2) i = 2 ͷ࣌ʜ ˜ n2 = x2 − (nT 1 x2) n1 , n2 = ˜ n2 ∥ ˜ n2 ∥ nT 1 n2 = nT 1 (x2 − (nT 1 x2) n1) 1 ∥ ˜ n2 ∥ = (nT 1 x2 − (nT 1 x2) nT 1 n1) 1 ∥ ˜ n2 ∥ = (nT 1 x2 − nT 1 x2) 1 ∥ ˜ n2 ∥ = 0 ΑΓ੒Γཱͭɽ جఈͳͷͰϊϧϜ͸
33. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ ೚ҙͷɹ࣍ݩͰಉ༷ʹ੒ཱ͢Δ͔Λ֬ೝ͢ΔͨΊʹ਺ֶతؼೲ๏ΛؤுΔʜ k (3) i = k − 1 ͷ࣌ʜ (4) i = k ͷ࣌ʜ ˜ nk−1 = xk−1 − k−2 ∑ j=1 (nT j xk−1) nj , nk−1 = ˜ nk−1 ∥ ˜ nk−1 ∥ ͕੒ΓཱͭͱԾఆ͢Δͱ ˜ nk = xk − k−1 ∑ j=1 (nT j xk) nj , nk = ˜ nk ∥ ˜ nk ∥ ͱॻ͚Δɽ͜Εʹ͍ͭͯ೚ҙͷجఈɹͱͷ಺ੵΛܭࢉ͢Δͱ ni
34. ෦෼ۭؒ άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ ೚ҙͷɹ࣍ݩͰಉ༷ʹ੒ཱ͢Δ͔Λ֬ೝ͢ΔͨΊʹ਺ֶతؼೲ๏ΛؤுΔʜ k (4) i = k ͷ࣌ʜ ˜ nk = xk − k−1 ∑ j=1 (nT j xk) nj , nk = ˜ nk ∥ ˜ nk ∥ ͱॻ͚Δɽ͜Εʹ͍ͭͯ೚ҙͷجఈɹͱͷ಺ੵΛܭࢉ͢Δͱ ni ͳͷͰɹɹɹ੒෼ͷΈ͕࢒Γ nT i nj = 0 (i ≠ j) i = j nT i nk = (nT i xk − (nT i xk) nT i ni) 1 ∥ ˜ nk ∥ = (nT i xk − nT i xk) 1 ∥ ˜ nk ∥ = 0 ͜Ε͕ɹɹɹɹɹɹͷશͯͷɹʹ͍ͭͯ੒ΓཱͭͷͰ
    άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ͸೚ҙ࣍ݩͰ࢖͑Δ 1 ≤ i ≤ k − 1 i
35. ෦෼ۭؒ ·ͱΊ Vd=3 E࣍ݩϕΫτϧۭؒͷ೚ҙͷϕΫτϧ͸
    Eݸͷجఈͷઢܗ݁߹ͰҰҙʹදݱͰ͖Δ Vd=3 x1 x3 x2 Wd=2 ਖ਼ن௚ަجఈ ௕͕͞Ͱ௚ަ͢ΔΑ͏ʹఆΊͨجఈ Vd=3 x1 x3 x2 Vd=3 n1 n3 n2 άϥϜʔγϡϛοτͷਖ਼ن௚ަԽ ༩͑ΒΕͨϕΫτϧ͔Β
    ਖ਼ن௚ަجఈΛ࡞Δ ෦෼ۭؒ ɹɹɹݸͷجఈͰදݱͰ͖ΔϕΫτϧۭؒ r( < d)
36. 065-*/& ·ͱΊ ओ੒෼෼ੳ ಛҟ஋෼ղ ෦෼ۭؒ ෦෼ۭؒ๏
37. 065-*/& ·ͱΊ ओ੒෼෼ੳ ಛҟ஋෼ղ ෦෼ۭؒ ෦෼ۭؒ๏
38. 065-*/& ओ੒෼෼ੳ ಋग़ͷྲྀΕͳͲ ຕ ຕ ຕ ϥάϥϯδϡͷະఆ৐਺๏ ݻ༗஋໰୊ ओ੒෼நग़ ຕ ·ͱΊ ຕ
39. ओ੒෼෼ੳ ಋग़ͷྲྀΕ ᶃ
    ֶशσʔλϕΫτϧ༝དྷͷߦྻɹΛ࡞Δ ᶇ
    ্ه໰୊Λϥάϥϯδϡͷະఆ৐਺๏Ͱղ͘ ᶈ
    ग़͖ͯͨݻ༗஋໰୊Λղ͘ ᶉ
    ओ੒෼Λநग़͢Δ ¯ X ᶄ
    ֶशσʔλߦྻɹʹ܎਺ϕΫτϧɹΛֻ͚ͨϕΫτϧɹΛ࡞Δ ¯ X aj sj ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } ᶆ
    ܎਺ϕΫτϧɹͷൣғΛࢦఆͯ͠෼ࢄɹɹɹͷ࠷େԽ໰୊Λઃఆ͢Δ Var{sj } aj
40. ओ੒෼෼ੳ ᶃ
    ֶशσʔλϕΫτϧ༝དྷͷߦྻɹΛ࡞Δ ¯ X X = xT 1 xT 2 ⋮ xT N = x11 x12 ⋯ x1d x21 x22 ⋯ x2d ⋮ ⋮ ⋱ ⋮ xN1 xN2 ⋯ xNd ¯ X = (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T = x11 − ¯ x1 x12 − ¯ x2 ⋯ x1d − ¯ xd x21 − ¯ x1 x22 − ¯ x2 ⋯ x2d − ¯ xd ⋮ ⋮ ⋱ ⋮ xN1 − ¯ x1 xN2 − ¯ x2 ⋯ xNd − ¯ xd
41. ओ੒෼෼ੳ ¯ Xaj = (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T aj1 aj2 ⋮ ajd = x11 − ¯ x1 x12 − ¯ x2 ⋯ x1d − ¯ xd x21 − ¯ x1 x22 − ¯ x2 ⋯ x2d − ¯ xd ⋮ ⋮ ⋱ ⋮ xN1 − ¯ x1 xN2 − ¯ x2 ⋯ xNd − ¯ xd aj1 aj2 ⋮ ajd ᶄ
    ֶशσʔλߦྻɹʹ܎਺ϕΫτϧɹΛֻ͚ͨϕΫτϧɹΛ࡞Δ ¯ X aj sj = (x11 − ¯ x1 )aj1 + (x12 − ¯ x2 )aj2 + ⋯ + (x1d − ¯ xd )ajd (x21 − ¯ x1 )aj1 + (x22 − ¯ x2 )aj2 + ⋯ + (x2d − ¯ xd )ajd ⋮ (xN1 − ¯ x1 )aj1 + (xN2 − ¯ x2 )aj2 + ⋯ + (xNd − ¯ xd )ajd = s1j s2j ⋮ sNj = sj ݸͷಛ௃ྔͷҰ࣍ؔ਺ͷܗ d
42. ओ੒෼෼ੳ ¯ XT ¯ X = (x1 − ¯ x x2 − ¯ x ⋯ xN − ¯ x) (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } = x11 − ¯ x1 x21 − ¯ x1 ⋯ xN1 − ¯ x1 x12 − ¯ x2 x22 − ¯ x2 ⋯ xN2 − ¯ x2 ⋮ ⋮ ⋱ ⋮ x1d − ¯ xd x2d − ¯ xd ⋯ xNd − ¯ xd x11 − ¯ x1 x12 − ¯ x2 ⋯ x1d − ¯ xd x21 − ¯ x1 x22 − ¯ x2 ⋯ x2d − ¯ xd ⋮ ⋮ ⋱ ⋮ xN1 − ¯ x1 xN2 − ¯ x2 ⋯ xNd − ¯ xd = ∑N i=1 (xi1 − ¯ x1 )2 ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ∑N i=1 (xi2 − ¯ x2 )2 ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ⋮ ⋮ ⋱ ⋮ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi2 − ¯ x2 )(xid − ¯ xd ) ⋯ ∑N i=1 (xid − ¯ xd )2 Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj
43. ओ੒෼෼ੳ ¯ XT ¯ X = (x1 − ¯ x x2 − ¯ x ⋯ xN − ¯ x) (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } = x11 − ¯ x1 x21 − ¯ x1 ⋯ xN1 − ¯ x1 x12 − ¯ x2 x22 − ¯ x2 ⋯ xN2 − ¯ x2 ⋮ ⋮ ⋱ ⋮ x1d − ¯ xd x2d − ¯ xd ⋯ xNd − ¯ xd x11 − ¯ x1 x12 − ¯ x2 ⋯ x1d − ¯ xd x21 − ¯ x1 x22 − ¯ x2 ⋯ x2d − ¯ xd ⋮ ⋮ ⋱ ⋮ xN1 − ¯ x1 xN2 − ¯ x2 ⋯ xNd − ¯ xd = ∑N i=1 (xi1 − ¯ x1 )2 ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ∑N i=1 (xi2 − ¯ x2 )2 ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ⋮ ⋮ ⋱ ⋮ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi2 − ¯ x2 )(xid − ¯ xd ) ⋯ ∑N i=1 (xid − ¯ xd )2 Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj
44. ओ੒෼෼ੳ ¯ XT ¯ X = (x1 − ¯ x x2 − ¯ x ⋯ xN − ¯ x) (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } = x11 − ¯ x1 x21 − ¯ x1 ⋯ xN1 − ¯ x1 x12 − ¯ x2 x22 − ¯ x2 ⋯ xN2 − ¯ x2 ⋮ ⋮ ⋱ ⋮ x1d − ¯ xd x2d − ¯ xd ⋯ xNd − ¯ xd x11 − ¯ x1 x12 − ¯ x2 ⋯ x1d − ¯ xd x21 − ¯ x1 x22 − ¯ x2 ⋯ x2d − ¯ xd ⋮ ⋮ ⋱ ⋮ xN1 − ¯ x1 xN2 − ¯ x2 ⋯ xNd − ¯ xd = ∑N i=1 (xi1 − ¯ x1 )2 ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ∑N i=1 (xi2 − ¯ x2 )2 ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ⋮ ⋮ ⋱ ⋮ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi2 − ¯ x2 )(xid − ¯ xd ) ⋯ ∑N i=1 (xid − ¯ xd )2 Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj
45. ओ੒෼෼ੳ ¯ XT ¯ X = (x1 − ¯ x x2 − ¯ x ⋯ xN − ¯ x) (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } = x11 − ¯ x1 x21 − ¯ x1 ⋯ xN1 − ¯ x1 x12 − ¯ x2 x22 − ¯ x2 ⋯ xN2 − ¯ x2 ⋮ ⋮ ⋱ ⋮ x1d − ¯ xd x2d − ¯ xd ⋯ xNd − ¯ xd x11 − ¯ x1 x12 − ¯ x2 ⋯ x1d − ¯ xd x21 − ¯ x1 x22 − ¯ x2 ⋯ x2d − ¯ xd ⋮ ⋮ ⋱ ⋮ xN1 − ¯ x1 xN2 − ¯ x2 ⋯ xNd − ¯ xd = ∑N i=1 (xi1 − ¯ x1 )2 ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ∑N i=1 (xi2 − ¯ x2 )2 ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ⋮ ⋮ ⋱ ⋮ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi2 − ¯ x2 )(xid − ¯ xd ) ⋯ ∑N i=1 (xid − ¯ xd )2 Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj
46. ओ੒෼෼ੳ ¯ XT ¯ X = (x1 − ¯ x x2 − ¯ x ⋯ xN − ¯ x) (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } = x11 − ¯ x1 x21 − ¯ x1 ⋯ xN1 − ¯ x1 x12 − ¯ x2 x22 − ¯ x2 ⋯ xN2 − ¯ x2 ⋮ ⋮ ⋱ ⋮ x1d − ¯ xd x2d − ¯ xd ⋯ xNd − ¯ xd x11 − ¯ x1 x12 − ¯ x2 ⋯ x1d − ¯ xd x21 − ¯ x1 x22 − ¯ x2 ⋯ x2d − ¯ xd ⋮ ⋮ ⋱ ⋮ xN1 − ¯ x1 xN2 − ¯ x2 ⋯ xNd − ¯ xd = ∑N i=1 (xi1 − ¯ x1 )2 ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ∑N i=1 (xi2 − ¯ x2 )2 ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ⋮ ⋮ ⋱ ⋮ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi2 − ¯ x2 )(xid − ¯ xd ) ⋯ ∑N i=1 (xid − ¯ xd )2 Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj
47. ओ੒෼෼ੳ ¯ XT ¯ X = (x1 − ¯ x x2 − ¯ x ⋯ xN − ¯ x) (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj = ∑N i=1 (xi1 − ¯ x1 )2 ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ∑N i=1 (xi2 − ¯ x2 )2 ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ⋮ ⋮ ⋱ ⋮ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi2 − ¯ x2 )(xid − ¯ xd ) ⋯ ∑N i=1 (xid − ¯ xd )2 = Nσ11 Nσ12 ⋯ Nσ1d Nσ12 Nσ22 ⋯ Nσ2d ⋮ ⋮ ⋱ ⋮ Nσ1d Nσ2d ⋯ Nσdd = NΣ ڞ෼ࢄߦྻ
48. ओ੒෼෼ੳ ¯ XT ¯ X = (x1 − ¯ x x2 − ¯ x ⋯ xN − ¯ x) (x1 − ¯ x)T (x2 − ¯ x)T ⋮ (xN − ¯ x)T ᶅ
    ϕΫτϧɹͷ෼ࢄɹɹɹΛܭࢉ͢Δ sj Var{sj } Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj = aT j Σaj = ∑N i=1 (xi1 − ¯ x1 )2 ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi1 − ¯ x1 )(xi2 − ¯ x2 ) ∑N i=1 (xi2 − ¯ x2 )2 ⋯ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ⋮ ⋮ ⋱ ⋮ ∑N i=1 (xi1 − ¯ x1 )(xid − ¯ xd ) ∑N i=1 (xi2 − ¯ x2 )(xid − ¯ xd ) ⋯ ∑N i=1 (xid − ¯ xd )2 = Nσ11 Nσ12 ⋯ Nσ1d Nσ12 Nσ22 ⋯ Nσ2d ⋮ ⋮ ⋱ ⋮ Nσ1d Nσ2d ⋯ Nσdd = NΣ aT j Σaj ڞ෼ࢄߦྻ ڞ෼ࢄߦྻͰஔ׵
49. ओ੒෼෼ੳ ᶆ
    ܎਺ϕΫτϧɹͷൣғΛࢦఆͯ͠෼ࢄɹɹɹͷ࠷େԽ໰୊Λઃఆ͢Δ Var{sj } aj ओ໰୊ maximize f(aj ) = Var{sj } = aT j Σaj subject to g(aj ) = ∥aj ∥2 − 1 = 0 ධՁؔ਺ʢ࠷େԽʣ ੍໿৚݅ ্هͷෆ౳੍ࣜ໿৚݅ԼͰධՁؔ਺Λղ͘ʢ࠷େԽ͢Δʣ͜ͱΛओ໰୊ͱ͢Δɽ ͜͜Ͱɼ੍໿৚݅ͱධՁؔ਺Λಉ࣌ʹຬͨ͢৽͍ؔ͠਺Λߟ͑Δͱʜ ϥάϥϯδϡؔ਺ ˜ L(aj , λ) = aT j Σaj − λ (∥aj ∥2 − 1) = aT j Σaj − λ (aT j aj − 1)
50. ओ੒෼෼ੳ ϥάϥϯδϡؔ਺ ˜ L(aj , λ) = aT j Σaj − λ (aT j aj − 1) ᶇ
    ্ه໰୊Λϥάϥϯδϡͷະఆ৐਺๏Ͱղ͘ ৚݅ᶃ ∂˜ L(aj , λ) ∂aj = (Σ + ΣT) aj − 2λaj = 2Σaj − 2λaj = 0 Σaj − λaj = (Σ − λ) aj = (Σ − λE) aj = 0 ࠷େԽͷͨΊͷඍ෼ɽݻ༗஋໰୊ͷܗʹͳΔʜ ͸ରশߦྻͳͷͰɹɹɹ͔Β Σ Σ = ΣT
51. ओ੒෼෼ੳ ϥάϥϯδϡؔ਺ ˜ L(aj , λ) = aT j Σaj − λ (aT j aj − 1) ᶇ
    ্ه໰୊Λϥάϥϯδϡͷະఆ৐਺๏Ͱղ͘ ৚݅ᶄ ∂˜ L(aj , λ) ∂λ = − (aT j aj − 1) = 0 aT j aj − 1 = 0 ੍໿৚݅ͦͷ΋ͷɽ
52. ओ੒෼෼ੳ ϥάϥϯδϡؔ਺ ˜ L(aj , λ) = aT j Σaj − λ (aT j aj − 1) ᶇ
    ্ه໰୊Λϥάϥϯδϡͷະఆ৐਺๏Ͱղ͘ ৚݅ᶅ λ ≥ 0 ϥάϥϯδϡະఆ৐਺ͷఆٛ
53. ओ੒෼෼ੳ ϥάϥϯδϡؔ਺ ˜ L(aj , λ) = aT j Σaj − λ (aT j aj − 1) ᶇ
    ্ه໰୊Λϥάϥϯδϡͷະఆ৐਺๏Ͱղ͘ ৚݅ᶆ λ (aT j aj − 1) = 0 ૬ิੑ৚݅ɽ
    ɹɹɹɹɹͳͷͰৗʹ੒Γཱͭʜ aT j aj − 1 = 0
54. ओ੒෼෼ੳ ϥάϥϯδϡؔ਺ ˜ L(aj , λ) = aT j Σaj − λ (aT j aj − 1) ᶇ
    ্ه໰୊Λϥάϥϯδϡͷະఆ৐਺๏Ͱղ͘ Ҏ্Λ·ͱΊͯ෼ࢄ࠷େԽ໰୊ʹ͓͚Δ,,5 ,BSVTI,VIO5VDLFS ৚݅ ᶃ (Σ − λE) aj = 0 ᶄ aT j aj − 1 = 0 ᶅ λ ≥ 0 ᶆ λ (aT j aj − 1) = 0 ͕ಘΒΕͨɽ͜Ε͸ɼᶃʹ͓͚Δݻ༗஋໰୊Λղ͘͜ͱͰղΛಘΔ͜ͱ͕Ͱ͖Δɽ (Σ − λE) aj = 0
55. ओ੒෼෼ੳ ᶈ
    ग़͖ͯͨݻ༗஋໰୊Λղ͘ ݻ༗஋໰୊ det (Σ − λE) = 0 a1 , a2 , . . . , ad λ1 ≥ λ2 ≥ . . . ≥ λd ߦྻ͕ࣜθϩʹͳΔ͜ͱΛ
    ׆༻ͯ͠ ݻ༗஋Λࢉग़ͯ͠
    େ͖͍ॱʹฒ΂Δ ݻ༗ϕΫτϧΛٻΊͯ
    ਖ਼نԽ͢Δ (Σ − λE) aj = 0
56. ओ੒෼෼ੳ ᶈ
    ग़͖ͯͨݻ༗஋໰୊Λղ͘ ݻ༗஋໰୊ ڞ෼ࢄߦྻɹ͕࣮ରশߦྻͰ͋Δ͜ͱ͔Βݻ༗ϕΫτϧɹ͸ͦΕͧΕ௚ަ͢Δɽ Σ a ࣮ରশߦྻͱݻ༗ϕΫτϧͷ௚ަੑ Σai = λi ai , Σaj = λj aj , (λi ≠ λj ) aT i (Σaj) = aT i Σaj = (ΣT ai) T aj = (Σai) T aj = (λi ai) T aj = λi aT i aj aT i (Σaj) = aT i (λj aj) = λj aT i aj λi aT i aj − λj aT i aj = 0 ⟺ aT i aj = 0 (λi ≠ λj ) ಉ͡΋ͷ (Σ − λE) aj = 0
57. ओ੒෼෼ੳ ᶈ
    ग़͖ͯͨݻ༗஋໰୊Λղ͘ ݻ༗஋໰୊ aT i aj = δij = { 1 (i = j) 0 (i ≠ j) ڞ෼ࢄߦྻɹ͕࣮ରশߦྻͰ͋Δ͜ͱ͔Βݻ༗ϕΫτϧɹ͸ͦΕͧΕ௚ަ͢Δɽ Σ a (Σ − λE) aj = 0 ΫϩωοΧʔͷσϧλ
58. ओ੒෼෼ੳ ᶈ
    ग़͖ͯͨݻ༗஋໰୊Λղ͘ ݻ༗஋໰୊ ڞ෼ࢄߦྻɹ͕࣮ରশߦྻͰ͋Δ͜ͱ͔Βݻ༗ϕΫτϧɹ͸ͦΕͧΕ௚ަ͢Δɽ Σ a ಉ͡Α͏ͳಛ௃ྔ͕͋ΓɼͦΕΒͷ෼ࢄ΍
    ڞ෼ࢄ͕ಉ͡Α͏ͳ஋ΛͱΔͱɹ͸ɹ࣍ݩ
    ΑΓ΋গͳ͍৘ใ͔࣋ͨ͠ͳ͘ͳΔɽ Σ = d × d Σ = d × k ϥϯΫམͪ (k ≤ d) d Σ (Σ − λE) aj = 0 ଞͱඃͬͯΔ
59. ओ੒෼෼ੳ (Σ − λE) aj = 0 ᶈ
    ग़͖ͯͨݻ༗஋໰୊Λղ͘ ݻ༗஋໰୊ ڞ෼ࢄߦྻɹ͕࣮ରশߦྻͰ͋Δ͜ͱ͔Βݻ༗ϕΫτϧɹ͸ͦΕͧΕ௚ަ͢Δɽ Σ a ඇθϩݻ༗஋͸࠷େͰɹݸಘΒΕΔ͕ɼ
    ੑ࣭ͷࣅͨಛ௃ྔͷ਺͚ͩڞ෼ࢄߦྻͷ
    ϥϯΫ͸མͪɼθϩʹͳΔݻ༗஋͕૿͑Δ Σ = d × k ϥϯΫམͪ (k ≤ d) ϥϯΫམͪͨ͠෼͚ͩ
    θϩʹͳΔݻ༗஋͕૿͑Δ λi = 0 ai = 0 Σ ≠ O ͳͷͰ d
60. ओ੒෼෼ੳ ᶉ
    ओ੒෼Λநग़͢Δ a1 , a2 , . . . , ad λ1 ≥ λ2 ≥ . . . ≥ λd ٻΊͨݻ༗஋ͱͦΕʹରԠ͢Δݻ༗ϕΫτϧ ֶशσʔλɹʹݻ༗ϕΫτϧɹΛֻ͚ͨϕΫτϧɹͷ෼ࢄΛߟ͑Δʜ ¯ X a1 s1 Var{s1 } = Var{ ¯ Xa1 } = aT 1 Σa1 = a1 (Σa1) = aT 1 (λ1 a1) = λ1 aT 1 a1 = λ1 ಉ༷ͷܭࢉ͕ଞͷݻ༗ϕΫτϧʹ͍ͭͯ΋੒ΓཱͭͷͰɼ
    ֶशσʔλɹͱݻ༗ϕΫτϧɹΛֻ͚ͨϕΫτϧɹ͕࠷େͷ
    ෼ࢄΛ࣋ͭͷ͸࠷େͷɹΛ࣋ͭɹΛ܎਺ͱͯ͠બΜͩͱ͖ɽ
    ͢ͳΘͪɼɹɹͷ࣌ɼ࠷େͷ෼ࢄɹΛ࣋ͭ௚ઢɹɹ͕Ҿ͚Δɽ ¯ X ai si λi ai i = 1 λ1 f1 (x) = aT 1 x = d ∑ i=1 a1i xi f1 (x)
61. ओ੒෼෼ੳ ᶉ
    ओ੒෼Λநग़͢Δ a1 , a2 , . . . , ad λ1 ≥ λ2 ≥ . . . ≥ λd ٻΊͨݻ༗஋ͱͦΕʹରԠ͢Δݻ༗ϕΫτϧ ֶशσʔλɹʹݻ༗ϕΫτϧɹΛֻ͚ͨϕΫτϧɹͷ෼ࢄΛߟ͑Δʜ ¯ X a1 s1 Var{s1 } = Var{ ¯ Xa1 } = aT 1 Σa1 = a1 (Σa1) = aT 1 (λ1 a1) = λ1 aT 1 a1 = λ1 ಉ༷ʹɹ͕େ͖͍ॱʹॏཁͳ৘ใΛ࣋ͭ௚ઢɹɹ͕Ҿ͚Δ λi fi (x) fj (x) = aT j x = d ∑ i=1 aji xi
62. ओ੒෼෼ੳ ᶉ
    ओ੒෼Λநग़͢Δ a1 , a2 , . . . , ad λ1 ≥ λ2 ≥ . . . ≥ λd ٻΊͨݻ༗஋ͱͦΕʹରԠ͢Δݻ༗ϕΫτϧ ͱ͜ΖͰɼશ෼ࢄྔɹɹΛߟ͑Δͱɼ Vtotal = d ∑ i=1 λi ͜ͷதͰݻ༗஋ɹΛҰͭऔΓग़ͯ͠શ෼ࢄྔʹ͓͚Δد༩཰ɹΛΈΔͱɼ
    ͦͷ੒෼͕ͲΕ͚ͩݩͷσʔλΛ࠶ݱ͍ͯ͠Δ͔Λ஌Δ͜ͱ͕Ͱ͖Δɽ ck = λk Vtotal Vtotal λi ck ݩͷσʔλͷ࠶ݱ཰
    Έ͍ͨͳ΋ͷʜ
63. ओ੒෼෼ੳ ᶉ
    ओ੒෼Λநग़͢Δ a1 , a2 , . . . , ad λ1 ≥ λ2 ≥ . . . ≥ λd ٻΊͨݻ༗஋ͱͦΕʹରԠ͢Δݻ༗ϕΫτϧ ͱ͜ΖͰɼશ෼ࢄྔɹɹΛߟ͑Δͱɼ Vtotal = d ∑ i=1 λi զʑ͕ཉ͍͠ͷ͸ྫ͑͹ɼݩͷσʔλͷΛ࠶ݱͰ͖Δ੒෼·ͰʂͳͲ
    ͳͷͰɼେ͖͍ݻ༗஋͔Βॱ൪ʹબΜͰશ෼ࢄྔ΁ͷྦྷੵد༩཰Λߟ͑Δ rk = k ∑ i=1 ci = ∑k i=1 λi Vtotal ≥ 0.8 Vtotal ɹதΛ௒͑Δ·ͰͷɹΛ࢖͏ Vtotal λi
64. ओ੒෼෼ੳ ·ͱΊ Var{sj } = 1 N sT j sj = 1 N ( ¯ Xaj) T ( ¯ Xaj) = 1 N aT j ¯ XT ¯ Xaj = aT j Σaj ˜ L(aj , λ) = aT j Σaj − λ (∥aj ∥2 − 1) Var{s1 } = Var{ ¯ Xa1 } = aT 1 Σa1 = a1 (Σa1) = aT 1 (λ1 a1) = λ1 aT 1 a1 = λ1 (Σ − λE) aj = 0 a1 , a2 , . . . , ad λ1 ≥ λ2 ≥ . . . ≥ λd ฏۉ஋Λج४ʹͨ͠σʔλͷҰ࣍ؔ਺ͷڞ෼ࢄߦྻΛݩͷڞ෼ࢄߦྻͰදݱ͢Δ ɹͷ஋Λ੍ݶͨ͠ϥάϥϯδϡؔ਺Λ࡞Δ ϥάϥϯδϡؔ਺͔Β༩͑ΒΕͨݻ༗஋໰୊Λղ͘ ٻΊΒΕͨݻ༗ϕΫτϧͰ࡞ΔҰ࣍ؔ਺͔Β෼ࢄΛܭࢉ͢ΔͱରԠ͢Δݻ༗஋ͱҰக ai ࠷େͷݻ༗஋ʹରԠ͢Δݻ༗ϕΫτϧ͕σʔλͷ෼ࢄΛ࠷େʹ͢Δ௚ઢʹҰக͢Δ
65. 065-*/& ·ͱΊ ओ੒෼෼ੳ ಛҟ஋෼ղ ෦෼ۭؒ ෦෼ۭؒ๏
66. 065-*/& ·ͱΊ ओ੒෼෼ੳ ಛҟ஋෼ղ ෦෼ۭؒ ෦෼ۭؒ๏
67. ಛҟ஋෼ղ 065-*/& ຕ ಛҟ஋෼ղ
68. ಛҟ஋෼ղ ಛҟ஋෼ղ X = UΛVT ·ͣ͸ఱԼΓతʹಛҟ஋෼ղΛఆٛ͠·͢ʜ ֤ߦྻͷҰൠతͳத਎Λ֬ೝ͢Δͱʜ U = (u1 , u2 , . . . , um) V = (v1 , v2 , . . . , vn) Λ = λ1 0 ⋯ 0 0 ⋯ 0 0 λ2 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ λp 0 ⋯ 0 0 0 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋯ 0 0 ⋯ 0 ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
69. ಛҟ஋෼ղ ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V (XT X) T = XT X ɹ͸ɹɹɹͷ࣮ߦྻͳͷͰɹɹ͸ɹɹɹͷ࣮ରশߦྻ XT X X m × n n × n
70. ಛҟ஋෼ղ X ೚ҙͷߦྻɹʹରͯ͠ʜ XT Xͷਖ਼نԽͨ͠ݻ༗ϕΫτϧΛɹɹɹɹɹɹɹɹͱ͢Δͱݻ༗஋໰୊Λղ͘͜ͱͰɹ V = (v1 , v2 , . . . , vn) XT Xvi = λi vi Λຬͨ͢ݻ༗஋ɹΛݻ༗ϕΫτϧɹʹରͯ͠ҰͭఆٛͰ͖Δɽ·ͨɼ λi vi vT i vj = δij = { 1 (i = j) 0 (i ≠ j) ΫϩωοΧʔͷσϧλ ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
71. ಛҟ஋෼ղ ݻ༗஋ɹΛਖ਼ͷ΋ͷͱθϩͷ΋ͷʹ෼͚Δ͜ͱ͕Ͱ͖Δ λi vi XT Xvi = vi (λi vi) = λi vT i vi = λi XT Xvi = λi vi λi = vT i XT Xvi = (Xvi) T Xvi = (Xvi) T Xvi = ∥Xvi ∥2 ≥ 0 ⟺ { λi ≠ 0 (1 ≤ i ≤ p) λi = 0 (p + 1 ≤ i ≤ n) ɹ͕େ͖͍ॱʹฒ΂ͯΔͷͰ
    ɹ͕ޙΖʹฒͿΑ͏ʹ࡞ΕΔ λi 0 ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V X ೚ҙͷߦྻɹʹରͯ͠ʜ XT Xͷਖ਼نԽͨ͠ݻ༗ϕΫτϧΛɹɹɹɹɹɹɹɹͱ͢Δͱݻ༗஋໰୊Λղ͘͜ͱͰɹ V = (v1 , v2 , . . . , vn)
72. ಛҟ஋෼ղ ࠓɼɹΛҎԼͷΑ͏ʹఆٛ͢Δͱʜ ui uT i uj = 1 λi λj (Xvi) T Xvj = 1 λi λj vT i XT Xvj = λj λi λj vT i vj = λj λi λj δij = δij = { 1 (i = j) 0 (i ≠ j) ͳͷͰɼɹɹɹɹɹɹ͸ޓ͍ʹ௚ަ͢Δਖ਼ن௚ަجఈʹͳΔʜ {u1 , u, . . . , um } ui = 1 λi Xvi ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
73. ಛҟ஋෼ղ ͳͷͰɼɹɹɹɹɹɹ͸ޓ͍ʹ௚ަ͢Δਖ਼ن௚ަجఈʹͳΔʜ {u1 , u, . . . , um } ͳͷͰߦྻɹ͸ U U = [u1 , u2 , . . . , up , up+1 , . . . , um] V = [v1 , v2 , . . . , vn] ·ͨɼߦྻɹ͸ V ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
74. ಛҟ஋෼ղ UTU = I VTV = I ߦྻɹͱɹ͸ͱ΋ʹਖ਼ن௚ަجఈ͔ΒͳΔͷͰ U V ͳͷͰɼɹɹɹɹɹɹ͸ޓ͍ʹ௚ަ͢Δਖ਼ن௚ަجఈʹͳΔʜ {u1 , u, . . . , um } ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
75. ಛҟ஋෼ղ UT XV = uT 1 uT 2 ⋮ um (Xv1 Xv2 ⋯ Xvn) = uT 1 Xv1 uT 1 Xv2 ⋯ uT 1 Xvn uT 2 Xv2 uT 2 Xv2 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvn ೚ҙͷߦྻɹʹରͯ͠ߦྻɹɼɹΛ࡞༻ͤ͞Δͱʜ U V X ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
76. ಛҟ஋෼ղ uT i Xvj = ( 1 λi Xvi) T Xvj = 1 λi vT i XT Xvj = λj λi vT i vj = λi δij = λi (i = j, 1 ≤ i ≤ p) 0 (i ≠ j) 0 (p + 1 ≤ i ≤ n) ٻΊͨߦྻͷҰ੒෼Λߟ͑Δͱʜ ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
77. ಛҟ஋෼ղ UT XV = uT 1 Xv1 uT 1 Xv2 ⋯ uT 1 Xvn uT 2 Xv2 uT 2 Xv2 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvn = uT 1 Xv1 uT 1 Xv2 ⋯ uT 1 Xvp uT 1 Xvp+1 ⋯ uT 1 Xvn uT 2 Xv1 uT 2 Xv2 ⋯ uT 2 Xvp uT 2 Xvp+1 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ uT p Xv1 uT p Xv2 ⋯ uT p Xvp uT p Xvp+1 ⋯ uT p Xvn uT p+1 Xv1 uT p+1 Xv2 ⋯ uT p+1 Xvp uT p+1 Xvp+1 ⋯ uT p+1 Xvn ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvp uT m Xvp+1 ⋯ uT m Xvn uT i Xvj = ( 1 λi Xvi) T Xvj = 1 λi vT i XT Xvj = λj λi vT i vj = λi δij = λi (i = j, 1 ≤ i ≤ p) 0 (i ≠ j) 0 (p + 1 ≤ i ≤ n) ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V
78. ಛҟ஋෼ղ UT XV = uT 1 Xv1 uT 1 Xv2 ⋯ uT 1 Xvn uT 2 Xv2 uT 2 Xv2 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvn = λ1 uT 1 Xv2 ⋯ uT 1 Xvp uT 1 Xvp+1 ⋯ uT 1 Xvn uT 2 Xv1 λ2 ⋯ uT 2 Xvp uT 2 Xvp+1 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ uT p Xv1 uT p Xv2 ⋯ λp uT p Xvp+1 ⋯ uT p Xvn uT p+1 Xv1 uT p+1 Xv2 ⋯ uT p+1 Xvp uT p+1 Xvp+1 ⋯ uT p+1 Xvn ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvp uT m Xvp+1 ⋯ uT m Xvn ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V uT i Xvj = ( 1 λi Xvi) T Xvj = 1 λi vT i XT Xvj = λj λi vT i vj = λi δij = λi (i = j, 1 ≤ i ≤ p) 0 (i ≠ j) 0 (p + 1 ≤ i ≤ n) (i = j, 1 ≤ i ≤ p) λi
79. ಛҟ஋෼ղ UT XV = uT 1 Xv1 uT 1 Xv2 ⋯ uT 1 Xvn uT 2 Xv2 uT 2 Xv2 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvn = λ1 0 ⋯ 0 0 ⋯ 0 0 λ2 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ λp 0 ⋯ 0 0 0 ⋯ 0 uT p+1 Xvp+1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V uT i Xvj = ( 1 λi Xvi) T Xvj = 1 λi vT i XT Xvj = λj λi vT i vj = λi δij = λi (i = j, 1 ≤ i ≤ p) 0 (i ≠ j) 0 (p + 1 ≤ i ≤ n) 0 (i ≠ j)
80. ಛҟ஋෼ղ UT XV = uT 1 Xv1 uT 1 Xv2 ⋯ uT 1 Xvn uT 2 Xv2 uT 2 Xv2 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvn = λ1 0 ⋯ 0 0 ⋯ 0 0 λ2 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ λp 0 ⋯ 0 0 0 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V uT i Xvj = ( 1 λi Xvi) T Xvj = 1 λi vT i XT Xvj = λj λi vT i vj = λi δij = λi (i = j, 1 ≤ i ≤ p) 0 (i ≠ j) 0 (p + 1 ≤ i ≤ n) 0 (p + 1 ≤ i ≤ n)
81. ಛҟ஋෼ղ ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V UT XV = uT 1 Xv1 uT 1 Xv2 ⋯ uT 1 Xvn uT 2 Xv2 uT 2 Xv2 ⋯ uT 2 Xvn ⋮ ⋮ ⋱ ⋮ uT m Xvm uT m Xv2 ⋯ uT m Xvn = λ1 0 ⋯ 0 0 ⋯ 0 0 λ2 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ λp 0 ⋯ 0 0 0 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 = Λ Λ uT i Xvj = ( 1 λi Xvi) T Xvj = 1 λi vT i XT Xvj = λj λi vT i vj = λi δij = λi (i = j, 1 ≤ i ≤ p) 0 (i ≠ j) 0 (p + 1 ≤ i ≤ n)
82. ಛҟ஋෼ղ UT XV = Λ ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V UUT XVVT = IXI = X = UΛVT X = UΛVT ɹͱɹ͸ਖ਼ن௚ަجఈ͔ΒͳΔߦྻͳͷͰʜ U V
83. ಛҟ஋෼ղ ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V X = UΛVT ͱ͜ΖͰɼڵຯ͕͋Δͷ͸ݻ༗஋͕ඇθϩͷ෦෼ͷΈͰ͋Δ͔Β X = Up Λp VT p = Up λ1 0 ⋯ 0 0 λ2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ λp VT p = ( λ1 u1 vT 1 λ2 u2 vT 2 ⋯ λp up vT p ) ͕ͨͬͯ͠ɼɹ͸ɹɹɹߦྻɼɹ͸ɹɹɹߦྻɼɹ͸ɹɹɹߦྻ
    ಛʹɹɹɹͷ࣌ɼ͸͡ύλ Q ಉ༷ʹ͓͘͜ͱ͕Ͱ͖Δɽɹ m × p p × p p × n n = p Up Λp VT p
84. ಛҟ஋෼ղ UΛq VT = U λ1 0 ⋯ 0 0 ⋯ 0 0 λ2 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ λq 0 ⋯ 0 0 0 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 VT = Xq ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V ɹͷத͔Βɹɹɹɹɹ·Ͱͷݻ༗஋Λબ୒͢Δͱɼ
    ɹɹҎ߱ͷ৘ใΛ࡟ݮͨ͠σʔλɹ͕ಘΒΕΔ λq (q < p) λi q + 1 Xq
85. ಛҟ஋෼ղ UΛq VT = U λ1 0 ⋯ 0 0 ⋯ 0 0 λ2 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ λq 0 ⋯ 0 0 0 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 VT = Xq ಛҟ஋෼ղ X = UΛVT ࣮ߦྻɹɹɹߦྻɹΛɹɹɹߦྻɹ ɹɹɹߦྻɹ ɹɹɹߦྻɹͷࡾ੒෼ʹ
    ෼ղ͢Δ͜ͱɽ m × n X m × m U m × n Λ n × n V ɹͷϥϯΫɹͷޡࠩ࠷খͱ͍͏ҙຯͰͷ࠷ྑۙࣅͱ͍͏ͷ͸͓ͦΒ͘ɼ
    ෼ࢄͷখ͍͞ํ޲ͷ৘ใ͔Β࡟͓ͬͯΓݩͷܗΛ࢒͍ͯ͠Δͱ͍͏ҙຯͰݴ͍ͬͯΔʁ X q
86. 065-*/& ·ͱΊ ओ੒෼෼ੳ ಛҟ஋෼ղ ෦෼ۭؒ ෦෼ۭؒ๏
87. ·ͱΊ ෦෼ۭؒ Vd=3 x1 x3 x2 Wd=2 ओ੒෼෼ੳ ˜ L(aj , λ) = aT j Σaj − λ (∥aj ∥2 − 1) (Σ − λE) aj = 0 a1 , a2 , . . . , ad λ1 ≥ λ2 ≥ . . . ≥ λd X = Up Λp VT p = Up λ1 0 ⋯ 0 0 λ2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ λp VT p = ( λ1 u1 vT 1 λ2 u2 vT 2 ⋯ λp up vT p ) ಛҟ஋෼ղ