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# 経済数学II 「第４章 線型モデルと行列代数」

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### 経済数学II 「第４章 線型モデルと行列代数」

1. 1. shito@seinan-gu.ac.jp 2020 5 1 • • • (1) 2 (2) (3) 2 非線形曲線 (nonlinear curve) 線型近似 (linear approximation) 1
2. 2. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 1 (Matrix and Vector) (1) (2) • 2 3x + y = 4 9x − 3y = 8 =⇒ 3 1 9 −3 x y = 4 8 3 • a11x1 + a12x2 + · · · + a1nxn = d1 a21x1 + a22x2 + · · · + a2nxn = d2 ... ... ... ... am1x1 + am2x2 + · · · + amnxn = dm (a) x (b) (c) A = x = d = II 2
3. 3. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 • A A = (aij) (i = 1, 2, · · · , m, j = 1, 2, · · · , n) (row) (column) • A = (aij) i j • × aij • m = n (3) • (column vector): :::::: 1 • (row vector): :::::: 1 II 3
4. 4. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 • 1 1 n ⇓ n n– n n– 1 • x d • x d • x 2×2 = x11 x12 x21 x22 =⇒ d m×1 =       d1 d2 ... dm       =⇒ (4) 1 Ax = d 2 (a) 2 A x (b) Ax d 1 ⇓ (Matrix Algebra) II 4
5. 5. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (1) pp.65–69 (2) 4.1 (p.69) 1 2 2 (1) A = (aij) B = (bij) A = B ⇐⇒ aij = bij for all i and j. 4 3 0 2 = 4 3 0 2 = 3 4 0 2 x y = 7 4 x = y = (2) • • 3 8 9 5 2×2 + 4 1 2 7 2×2 =    a11 a12 a21 a22 a31 a32    3×2 −    b11 b12 b21 b22 b31 b32    3×2 = (3) =⇒ 7 3 −1 0 5 = 1 2 a11 a12 a21 a22 = II 5
6. 6. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (4) • AB a11 a12 a21 a22    b11 b12 b21 b22 b31 b32    = a11 a12 a21 a22 b11 b12 b13 b21 b22 b23 = 4 3 2 5 3 2 = 9 3 2 7 8 3 2 4 1 5 = 4 3 5    2 9 6    = (5) (6) : x1 + x2 + x3 = 3 j=1 xj Quiz (a) 5 i=2 yi = (b) x0 + x1 + x2 + x3 = (c) 3 j=1 ajxj = (d) n i=0 aixi = (e) 3 k=1 axk = a 3 k=1 xk II 6
7. 7. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 • xi i xi • a11 a12 a21 a22 b11 b12 b21 b22 = c11 c12 c21 c22 cij = 2 k=1 aikbkj (i = 1, 2, j = 1, 2) c11 = c12 = c21 = c22 = (1) pp.70–78 (2) 4.2 1–7 II 7
8. 8. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 3 (1) u =    4 2 5    v =    2 1 2    u v = uv = (2) 2 n n (a) u = 3 2 or u = ( 3 2 ) u u (0, 0) u or u 2 (3, 2) 1 u II 8
9. 9. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (b) (c) v = 1 4 u = 3 2 u + v = v − u = II 9
10. 10. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (d) 1 1 3v + 2u = 3 1 4 + 2 3 2 = 9 16 1 2 n i=1 kivi = k1v1 + k2v2 + · · · + knvn ki vi (3) 1 (linearly dependent) 1 (linearly independent) 1 v1, · · · , vn 1 1 1 1 3 1 v1 = 2 7 v2 = 1 8 v3 = 4 5 1 1 3v1 − 2v2 = v3 ↔ 3v1 − 2v2 − v3 = 0 0 ( 0 0 ) 1 1 2 m v1, · · · , vn n i=0 kivi = 0m×1 1 k1, · · · , kn 1 i ki = 0 1 II 10
11. 11. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (4) 1 v1 = 3 2 v2 = 6 4 v1 = 3 2 v2 = −3 −2 2 =⇒ (5) 1 v1 = 1 4 , v2 = 3 2 II 11
12. 12. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 v1 v2 1 ⇓ 2 2 1 1 ⇓ 2 3 1 1 (6) (a) 2 u v 1 2 • 1 • 2 • 2 1 u v 2 2 1 • 1 2 1 • 2 1 2 2 2 2 II 12
13. 13. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 • ( 1 0 ) ( 0 1 ) i 1 0 (b) 3 3 3 3 p.87 4.4 (c) n- n- n- n- 1 n n (1) pp.79–88 (2) 4.3 1–6 (p.89) 4 a b A B : a + b = b + a : ab = ba : (a + b) + c = a + (b + c) : (ab)c = a(bc) : a(b + c) = ab + ac (1) pp.90–93 (2) 4.4 1–5 II 13
14. 14. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 5 (1) (Identity Matrices) 1 I 2×2 = I 3×3 = I 4×4 = • 1 1 × a = a × 1 = a IA = AI = A A 2×3 = 1 2 3 2 0 3 I 2×2 A 2×3 = A 2×3 I 3×3 = • A m×n I n×n B n×p = • (I)2 = II 14
15. 15. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (2) (Null Matrices) O = 0 0 0 0 0 0 • A m×n + O m×n = O m×n + A m×n = A m×n • A m×n O n×p = O m×p O q×m A m×n = O q×n (3) • ab = 0 a b AB = 2 4 1 2 −2 4 1 −2 = • cd = ce d = e c = 0 C = 2 3 6 9 , D = 1 1 1 2 , E = −2 1 3 2 CD = CE = (1) pp.94–96 (2) 4.5 1–3 6 (1) (transposed matrices) A = 2 5 4 9 A = AT = B m×n B n×m II 15
16. 16. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (2) (A ) = (A + B) = (AB) = (3) (inverse matrices) • A • AA−1 = A−1 A = • A A ⇐⇒ A ⇐⇒ A • (A−1 )−1 = A • A n × n A−1 n × n • • (AB)−1 = • (A )−1 = II 16
17. 17. www.seinan-gu.ac.jp/˜shito 2020 5 1 18:00 (4) A (n×n) x (n×1) = d (n×1) (1) pp.97–103 (2) 4.6 1–5 II 17
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