Logseq/Major Study/pages/hls__5장_행렬식_1788006854821_0.md

file:: 5장_행렬식_1788006854821_0.pdf file-path:: ../assets/5장_행렬식_1788006854821_0.pdf

• 두 행을 교환하면 행렬식의 부호가 바뀐다 (두 열을 교환하는 경우도 마찬가지이다) ls-type:: annotation hl-page:: 3 hl-color:: yellow id:: 6a92d460-daa4-4daf-9de2-95975438dbac
• detA = 0은 행렬 A의 열들이 선형독립이 아님을 의미한다 ls-type:: annotation hl-page:: 3 hl-color:: yellow id:: 6a92d4d8-c967-4b78-95ef-3b71f96a4f84
• 모든 치환행렬의 행렬식은 1 또는 -1이다 ls-type:: annotation hl-page:: 3 hl-color:: yellow id:: 6a92d502-2f6f-472d-894a-5ac8651abc7c
• 행 교환은 detA에 -1을 곱하는 것 ls-type:: annotation hl-page:: 4 hl-color:: yellow id:: 6a92d52c-03a3-43af-8994-081bde3286d8
• 행렬 A의 행렬식이 각 행에 대해 선형적이라는 것이다 ls-type:: annotation hl-page:: 4 hl-color:: yellow id:: 6a92d555-5dab-4dc6-a08d-9f06288f4b03
• [:span] ls-type:: annotation hl-page:: 4 hl-color:: yellow id:: 6a92d586-1f49-4ac6-9702-4c9ff4d23a9e hl-type:: area hl-stamp:: 1788007813121
• 식 (2)에는 또 하나의 '수수께끼 같은 -1(음수 부호)'이 있다. ls-type:: annotation hl-page:: 5 hl-color:: yellow id:: 6a92d733-c351-493c-91d7-8103ec135192
• [:span] ls-type:: annotation hl-page:: 6 hl-color:: yellow id:: 6a92d780-e910-47e4-8e07-9de8ee47efb8 hl-type:: area hl-stamp:: 1788008319319
• [:span] ls-type:: annotation hl-page:: 6 hl-color:: yellow id:: 6a92d79b-46e1-4451-809d-9e8780197e46 hl-type:: area hl-stamp:: 1788008346577
• 이때 행렬 A = [a b; c d]의 여인수는 여인수행렬(cofactor matrix) C = [d -c; -b a] 안에 있다 ls-type:: annotation hl-page:: 6 hl-color:: yellow id:: 6a92d858-f867-4553-a090-6dff09b0a721
• 놀라운 사실은 A와 Cᵀ의 곱은 detA와 I의 곱과 같다는 것이다. ls-type:: annotation hl-page:: 6 hl-color:: yellow id:: 6a92d85e-68f5-4e72-ba9b-31f54593f0e8
• [:span] ls-type:: annotation hl-page:: 7 hl-color:: yellow id:: 6a92d867-4d38-4204-8f38-6f97d078f2a2 hl-type:: area hl-stamp:: 1788008549857
• [:span] ls-type:: annotation hl-page:: 7 hl-color:: yellow id:: 6a92d871-f906-4d54-9939-214b2138c412 hl-type:: area hl-stamp:: 1788008560529
• 행렬 Cᵀ는 detA로 나누어야 한다. 역행렬 A⁻¹의 모든 성분은 두 행렬식의 비율, 즉 (Cᵀ 안의 크기 (n-1)인 여인수) / (크기 n인 detA)이다 ls-type:: annotation hl-page:: 7 hl-color:: yellow id:: 6a92d953-45be-4c63-8a8b-e745adc17be4
• 행렬식이 1인 경우이다. 따라서 A⁻¹는 정확히 Cᵀ = 수반행렬(adjugate matrix) ls-type:: annotation hl-page:: 7 hl-color:: yellow id:: 6a92d974-f8ff-49ad-92b6-194f7fed92b7
• [:span] ls-type:: annotation hl-page:: 8 hl-color:: yellow id:: 6a92d9bf-5360-41c6-a359-f7ba5a34fc0d hl-type:: area hl-stamp:: 1788008893880
• [:span] ls-type:: annotation hl-page:: 8 hl-color:: yellow id:: 6a92d9c8-226a-418d-89d3-304dee2950a1 hl-type:: area hl-stamp:: 1788008903554
• [:span] ls-type:: annotation hl-page:: 5 hl-color:: yellow id:: 6a931640-8a00-492e-9be1-f8b712b408ed hl-type:: area hl-stamp:: 1788024382709
• [:span] ls-type:: annotation hl-page:: 5 hl-color:: yellow id:: 6a931709-11a2-469a-aa77-a7346c1ead5c hl-type:: area hl-stamp:: 1788024584039
• [:span] ls-type:: annotation hl-page:: 7 hl-color:: yellow id:: 6a93171f-9088-4ecf-bb2e-4453d75beed4 hl-type:: area hl-stamp:: 1788024605547
• 모든 n에 대해 detAₙ = n+1 ls-type:: annotation hl-page:: 8 hl-color:: yellow id:: 6a93173e-bc93-4d9b-b218-aac33dee3dde
• 가역행렬 A는 detA ≠ 0 ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a9439f3-8c77-4ce4-bf40-de233cd12495
• 특이행렬 A는 detA = 0 ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a943a00-fc7a-40ae-8856-7a6371a40c25
• Ax = λx에서 고윳값 λ와 고유벡터 x는 고윳값 방정식 (A−λI)x = 0을 만족한다. 즉 A−λI가 특이행렬이고 det(A−λI) = 0 ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a943a60-ddf1-47f7-93fb-85ab1f858a28
• 행렬이 삼각행렬이나 대각행렬이라면, 다음과 같이 단순히 주대각성분들을 곱함으로써 행렬식을 계산 ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a943a8d-513d-4a41-842f-dc51c4309ade
• [:span] ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a943a93-ffe5-493c-beec-c2e373dfec63 hl-type:: area hl-stamp:: 1788099218739
• [:span] ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a943a9f-5ef4-4a1a-aeae-6b7556fdeb58 hl-type:: area hl-stamp:: 1788099231645
• [:span] ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a943aa6-c986-4f50-b6ae-fa2af53fff53 hl-type:: area hl-stamp:: 1788099238051
• 직교행렬 Q의 행렬식은 1 또는 -1이다. ls-type:: annotation hl-page:: 11 hl-color:: yellow id:: 6a943ace-dea6-4c62-bf04-eb09243c01ef
• [:span] ls-type:: annotation hl-page:: 12 hl-color:: yellow id:: 6a943b00-83df-4cd4-b047-1f23ac2d8d6a hl-type:: area hl-stamp:: 1788099328536
• A = LU이면 detA = (detL)(detU) = detU이다. 삼각행렬 U의 행렬식은 피벗들의 곱 ls-type:: annotation hl-page:: 12 hl-color:: yellow id:: 6a943b6f-c65a-4e0b-b6e8-afae671d15f7
• PA = LU이면 행 교환이 존재하므로 detP = 1 또는 -1이다(치환행렬 ls-type:: annotation hl-page:: 12 hl-color:: yellow id:: 6a943b74-de8b-439f-80f0-6a3e074bc78b
• [:span] ls-type:: annotation hl-page:: 12 hl-color:: yellow id:: 6a943b90-0266-4228-baf4-227d765daec4 hl-type:: area hl-stamp:: 1788099472392
• 행렬 A의 두 행이 동일하면 A의 행렬식은 0이다 ls-type:: annotation hl-page:: 12 hl-color:: yellow id:: 6a943be0-7639-45fc-ae5e-fa811c704e4f
• j행에서 i행에 d를 곱한 것을 빼도 detA는 변하지 않는다 ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a943c8c-83fb-4053-afa3-f1a2b63d5179
• [:span] ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a943c91-84b2-4cc8-86de-b1018447eac9 hl-type:: area hl-stamp:: 1788099729531
• (행 교환이 없는) 소거 과정은 행렬식을 바꾸지 않고 A를 간단한 형태로 만든다 ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a943cf9-6d55-4024-ba3b-d3a3996e7755
• [:span] ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a943d04-373b-4865-b6d7-26b3b6f77324 hl-type:: area hl-stamp:: 1788099843701
• [:span] ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a943d32-0ad3-46e3-bc58-5936bbb3736c hl-type:: area hl-stamp:: 1788099890542
• 항등행렬 I의 1열을 x로 바꾼다. 이렇게 바꾼 삼각행렬 M₁의 행렬식은 x₁이다. M₁에 행렬 A를 곱하면 1열은 Ax가 되고, 가정에 따라 Ax = b이다. B₁의 나머지 열은 행렬 A와 동일하다. ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a943dc0-8127-417c-9142-7f364dec2cbc
• [:span] ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a943dc5-2624-4f08-b3e4-b2f6a1d99dc2 hl-type:: area hl-stamp:: 1788100037189
• [:span] ls-type:: annotation hl-page:: 14 hl-color:: yellow id:: 6a943dfb-3873-4e19-a53f-6bd6e3d16dc2 hl-type:: area hl-stamp:: 1788100091108
• [:span] ls-type:: annotation hl-page:: 14 hl-color:: yellow id:: 6a943e05-9a21-41e1-b292-8eb0ff3c1eb4 hl-type:: area hl-stamp:: 1788100100768
• [:span] ls-type:: annotation hl-page:: 14 hl-color:: yellow id:: 6a943e84-a662-40ad-9401-d69a0546bb42 hl-type:: area hl-stamp:: 1788100227948
• [:span] ls-type:: annotation hl-page:: 15 hl-color:: yellow id:: 6a943ef3-97cd-4c00-9e44-307fc49d3168 hl-type:: area hl-stamp:: 1788100339170
• 크레이머 규칙에서 각 Bⱼ의 행렬식은 행렬 A의 여인수이다. ls-type:: annotation hl-page:: 15 hl-color:: yellow id:: 6a943efb-271f-4733-ab93-21fa9bd6557c
• A⁻¹는 행렬 A의 여인수들을 포함한다 ls-type:: annotation hl-page:: 15 hl-color:: yellow id:: 6a943f06-f385-4965-81c1-6706552d7c05
• D(A) = detAB/detB가 앞서 살펴본 성질 ❶, ❷, ❸을 만족함을 보이고자 한다 ls-type:: annotation hl-page:: 13 hl-color:: yellow id:: 6a94462d-4af3-413d-b22b-6911e58b9b2c
• [:span] ls-type:: annotation hl-page:: 14 hl-color:: yellow id:: 6a94464a-ed55-45c7-96d4-135fa44a9387 hl-type:: area hl-stamp:: 1788102217715
• 마지막 4개의 행렬식의 값은 각각 d, -c, -b, a이다(이들은 모두 여인수이다) ls-type:: annotation hl-page:: 15 hl-color:: yellow id:: 6a944666-5084-47c1-b9a8-0960750738ea
• [:span] ls-type:: annotation hl-page:: 16 hl-color:: yellow id:: 6a944680-4b3c-4e0c-8151-af9190aef983 hl-type:: area hl-stamp:: 1788102271902
• [:span] ls-type:: annotation hl-page:: 16 hl-color:: yellow id:: 6a944693-c4eb-4ff1-95d6-796a200430e3 hl-type:: area hl-stamp:: 1788102291398