6-9习题

Ý Ø16-9Ù;.SKù) ~~~1. ® A =  1 0 i0 1 2 −i 2 5 §Ù¥ i = √−1§©O¦ ‖A‖1§‖A‖2§ ‖A‖∞Ú ‖A‖F" y²µ ‖A‖1 = max 1≤j≤3 3∑ i=1 |aij| = max{2, 3, 8} = 8 ‖A‖2 = √ λmax(AHA) = 6 ‖A‖∞ = max 1≤i≤3 n∑ j=1 |aij| = max{2, 3, 8} = 8 ‖A‖F = √∑ |aij|2 = √ 37 ~~~2. � A ´ n �Œ_Ý §…k x∗ § Ax = b �°()§xˆ ´ Ax = b �Cq )§r = b− Axˆ"y² ‖x∗ − xˆ‖ x∗ ≤ κ(A)‖r‖‖b‖ Ù¥ κ(A) = ‖A‖‖A−1‖§¿…•þ‰ê†Ý ‰êƒN" y²µdu Ax∗ − Axˆ = A(x∗ − xˆ) = r Kk x∗ − xˆ = A−1r � ‖x∗ − xˆ‖ ≤ ‖A−1‖ · ‖r‖ Ӟ§Ï Ax∗ = b§k ‖b‖ ≤ ‖A‖‖x∗‖ Ïdkeª¤á ‖x∗ − xˆ‖ · ‖b‖ ≤ ‖A−1‖ · ‖r‖ · ‖A‖ · ‖x∗‖ = ‖x∗ − xˆ‖ ‖x∗‖ ≤ κ(A) · ‖r‖ ‖b‖ ~~~3. ® A = 2 1 00 0 1 0 1 0 §¦ eAtÚ sin(At)" )µN´¦� P =  1 −1 1−3 1 0 3 1 0  , P−1 = 1 6 0 −1 10 3 3 6 4 2  ¦� P−1AP = −1 1 2  u´N´OŽ� eAt = P e−t et e2t P−1 = 1 6 6e2t 4e2t − 3et − e−t 2e2t − 3et + e−t0 3et + 3e−t 3et − 3e−t 0 3et − 3e−t 3et + 3e−t  Ú sin(At) = P sin−t sin t sin 2t P−1 = 1 6 sin 2t 4 sin 2t− 2 sin t 2 sin 2t− 4 sin t0 0 6 sin t 0 6 sin t 0  Œ^,˜«ª¦) )µN´¦� det(λI − A) = (λ+ 1)(λ− 1)(λ− 2)"� g(λ) = c0 + c1λ+ c2λ 2 Kd  g(−1) = c0 − c1 + c2 = e−t g(1) = c0 + c1 + c2 = e t g(2) = c0 + 2c2 + 4c2 = e 2t )�  c0 = 1 3(3e t + e−t − e2t) c1 = 1 2(e t − e−t) c2 = 1 6(e −t − 3et + 2e2t) l eAt = c0I + c1A+ c2A 2 = 1 6 6e2t 4e2t − 3et − e−t 2e2t − 3et + e−t0 3et + 3e−t 3et − 3e−t 0 3et − 3e−t 3et + 3e−t  d  g(−1) = c0 − c1 + c2 = sin−t g(1) = c0 + c1 + c2 = sin t g(2) = c0 + 2c2 + 4c2 = sin 2t ¦�  c0 = −13(sin 2t− 2 sin t) c1 = sin t c2 = 1 3(sin 2t− 2 sin t) Ïdk sinAt = c0I + c1A+ c2A 2 = 1 6 sin 2t 4 sin 2t− 2 sin t 2 sin 2t− 4 sin t0 0 6 sin t 0 6 sin t 0  ~~~4. � a = [a1 a2 a3 a4]T§x = (xij)2×4 ,¦ d(xa)dx Ú d(xa)T dx . )µN´�� xa =  4∑ k=1 x1kak 4∑ k=1 x2kak  Ï k dxa dx = [ ∂(xa) ∂x11 ∂(xa) ∂x12 ∂(xa) ∂x13 ∂(xa) ∂x14 ∂(xa) ∂x21 ∂(xa) ∂x22 ∂(xa) ∂x23 ∂(xa) ∂x24 ] =  a1 a2 a3 a4 0 0 0 0 0 0 0 0 a1 a2 a3 a4  Ú d(xa)T dx = [ ∂(xa)T ∂x11 ∂(xa)T ∂x12 ∂(xa)T ∂x13 ∂(xa)T ∂x14 ∂(xa)T ∂x21 ∂(xa)T ∂x23 ∂(xa)T ∂x23 ∂(xa)T ∂x24 ] = [ a1 0 a2 0 a3 0 a4 0 0 a1 0 a2 0 a3 0 a4 ] ~~~5. y² H+ = H �¿‡^‡´ H2´˜� HermiteÝ §…k rankH2 = rankH y²µXJ H+ = H Kk (H2)2 = H4 = H3 ·H = H2 (H2)H = (HH+)H = H2 Ïd H2´˜� HermiteÝ "Ӟ rankH = rankH3 ≤ rankH2 ≤ rankH Ïdk rankH2 = rankH"‡ƒXJ (H2)H = H2§Kk H ∈ H{3, 4}§qϏ rankH2 = rankH§l 3˜‡ U ¦� H = H2U§Ï k H3 = H2 ·H = H2 ·H2 · U = (H2)2 · U = H2 · U = H = H ∈ H{1, 2}� H ∈ H{1, 2, 3, 4}§= H = H+" ~~~6. ®Ý A =  1 0 −1 10 2 2 2 −1 4 5 3  , b =  4−2 −2  1.^2Â_Ý {�½y1§| Ax = b´Ä´ƒN� 2. ¦ Ax = b�4�‰ê)½ö4�‰ê��¦)£Ñ)�a.¤ )µN´¦� A�÷©) A = PQ =  1 00 2 −1 4 [1 1 −1 1 0 1 1 1 ] u´§Œ±¦Ñ A�2Â_Ý  A+ = QH(QQH)−1(PHP )−1PH =  1 0 0 1 −1 1 1 1 [3 00 3 ]−1 [ 2 −4 −4 20 ] [ 1 0 −1 0 2 4 ] = 1 18  5 2 −1 1 1 1 −4 −1 2 6 3 0  du AA+b = A  1 0 −1 1  =  30 −3  6= b ¤±§ Ax = b´ØƒN�§§�4�‰ê��¦) x0 = A +b = [1 0 − 1 1]T ~~~7. b A ∈ Cn×n� n‡‚5Ã'�A�•þ x1, x2, . . . , xn§y²µ A ⊗ Ak n2‡ ‚5Ã'�A�•þ§¿r¦‚�EÑ5" y²µ� Axi = λixi, (i = 1, 2, . . . , n)§Ï (A⊗ A)(xi ⊗ xj) = (Axi)⊗ (Axj) = (λixi)⊗ (λjxj) = (λiλj)(xi ⊗ xj) ¤± xi ⊗ xj (i, j = 1, 2, . . . , n)Ñ´ A⊗ A�A�•þ"e¡y²§‚´‚5Ã'�" �3˜|Iþ ki,j (i, j = 1, 2, . . . , n)§¦� n∑ i=1 n∑ j=1 kij(xi ⊗ xj) = 0 = n∑ j=1 (( n∑ i=1 kijxi )⊗ xj) = 0 2� xi = [ ξ1i, ξ2i, . . . , ξni ] T§Kk n∑ j=1 ( n∑ i=1 kijξli ) xj = 0(l = 1, 2, . . . , n) Ϗ x1, x2, . . . , xn´‚5Ã'�§¤± n∑ i=1 kijξli = 0, (l, j = 1, 2, . . . , , n) Ò´ n∑ i=1 kijxi = 0, (j = 1, 2, . . . , n) 2d x1, x2, . . . , xn´‚5Ã'�§Œ� kij = 0, (i, j = 1, 2, . . . , n)"Ïd xi⊗xj, (i, j = 1, 2 . . . , n)´ A⊗ A� n2‡‚5Ã'�A�•þ" ~~~8. ¦)Ý § AX +XB = F Ù¥ A = [ 1 −1 0 2 ] , B = [−3 4 0 −1 ] , F = [ 0 5 2 −9 ] . )µ A⊗ I2 + I2 ⊗BT =  −2 0 −1 0 4 0 0 −1 0 0 −1 0 0 0 4 1  ӞŒ±��Ý F �1.† vec(F ) = [ 0 5 2 − 9 ]T ¦)Ý § (A⊗ I2 + I2 ⊗BT )vec(X) = vec(F ) �Xe�) x1 = 1, x2 = k, x3 = −2, x4 = −1 Ù¥ k?¿�~ê§� Ý §�Ï) X = [ 1 0 −2 −1 ] + k [ 0 1 0 0 ] Ù¥ k?¿�~ê"