第 2 章 陣列(Array)與矩陣(Matrix)的運算

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1 第 2 章 陣列(Array)與矩陣(Matrix)的運算
2019/4/26 Puff! The magic dragon, live by the sea…

2 2019/4/26 Puff! The magic dragon, live by the sea…

3 列向量 & 行向量 Row vector 列向量 [5, 7, 2]; [3, 4, -6, 7] Column vector 行向量
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4 陣 列 Arrays 2019/4/26 Puff! The magic dragon, live by the sea…

5 轉置 transpose 2019/4/26 Puff! The magic dragon, live by the sea…

6 Transpose matrix g=[1;2;3;4]; g1=g’; or g=[1 2 3 4];
2019/4/26 Puff! The magic dragon, live by the sea…

7 Transpose matrix k=[1 -9 1 0 3; -4 4 1 1 2; 24 2 2 36 2] or
k1=k’ 2019/4/26 Puff! The magic dragon, live by the sea…

8 產生等間距大向量 x=[m:q:n] ?x=[0:2:8] x = 0 2 4 6 8 ?y=[-3:2] y =
?y=[-3:2] y = 2019/4/26 Puff! The magic dragon, live by the sea…

9 產生等間距大向量 ?z=[0:0.1:2] z = Columns 1 through 7
Columns 8 through 14 Columns 15 through 21 2019/4/26 Puff! The magic dragon, live by the sea…

10 產生等間距大向量 ?zzz=[3:-0.5:0] zzz =
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11 linespace(x1,x2,n) 產生等間距列向量
the same as [5:0.1:8] n 表示列向量內含的元素個數 如果省略則內定值為1 2019/4/26 Puff! The magic dragon, live by the sea…

12 logspace(x1,x2,n) 產生具有對數等間距的向量
logspace (a, b, n) 在10a~10b間(含)取的點數 logspace(-1,1,4)=[0.1, ,2.1544, 10.0] 2019/4/26 Puff! The magic dragon, live by the sea…

13 2-Dimensional Arrays Puff! The magic dragon, live by the sea…
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14 合併 矩陣(行數增加) 矩陣A與r需具有相同的列數 ?A=[2,4,10; 16,3,7] A = 2 4 10 16 3 7
?r=[1,1;0,0] r = ?B=[A r] B = 矩陣A與r需具有相同的列數 2019/4/26 Puff! The magic dragon, live by the sea…

15 合併 矩陣(列數增加) 矩陣A與r需具有相同的行數 ?A=[2,4,10; 16,3,7] A = 2 4 10 16 3 7
?r=[1,2,3] r = ?B=[A; r] B = 矩陣A與r需具有相同的行數 2019/4/26 Puff! The magic dragon, live by the sea…

16 Transpose matrix (complex conjugate)
?A=[3+2*i, 2+6*i; 5+4*i, 3+5*i] A = i i i i ?A' ans = i i i i 2019/4/26 Puff! The magic dragon, live by the sea…

17 Array addressing定址 修改矩陣內元素 D(1,3)=6 B = 2 4 10 16 3 7 1 2 3 ?B(2,2)=0
?B(2,2)=0 2019/4/26 Puff! The magic dragon, live by the sea…

18 Array addressing定址 2. V(:) V向量內所有元素 ?V=[1 2 3 4 5] V = 1 2 3 4 5 ?V(:)
?V(:) ans = 1 2 3 4 5 2019/4/26 Puff! The magic dragon, live by the sea…

19 Array addressing定址 3. V(2:4) V(2), V(3), V(4), V(5), V(6) V =
?V(2:4) ans = 2019/4/26 Puff! The magic dragon, live by the sea…

20 Array addressing定址 A(:,3) 矩陣A第三行所有的元素 A = 3 5 11 17 1 8 2 3 4 ?A(:,3)
?A(:,3) ans = 11 8 4 2019/4/26 Puff! The magic dragon, live by the sea…

21 Array addressing定址 5. A(:,2:3) 矩陣A第2行至第3行所有的元素 A = 3 5 11 17 1 8 2 3 4
?A(:,2:3) ans = 2019/4/26 Puff! The magic dragon, live by the sea…

22 Array addressing定址 6. A(2:3,1:3)矩陣A第2至第3列,第1至第3行所有的元素 (可用以分割矩陣) A =
?A(2:3,2:3) ans = 2019/4/26 Puff! The magic dragon, live by the sea…

23 ?B=A(:,5:-1:1) %把A陣列行排列的次序顛倒過來 B = 3 0 4 9 6 0 0 7 5 1
?A(1,5)=3 ?B=A(:,5:-1:1) %把A陣列行排列的次序顛倒過來 B = 2019/4/26 Puff! The magic dragon, live by the sea…

24 B = C = ?B(2,:)=C 2019/4/26 Puff! The magic dragon, live by the sea…

25 ?E=D([2,2,2,1],:) %將陣列D的第二列重複三次,第一列重複一次 E =
?E=D([2,2,2,1],:) %將陣列D的第二列重複三次,第一列重複一次 E = 2019/4/26 Puff! The magic dragon, live by the sea…

26 Using ‘clear’ to avoid errors
clear A 2019/4/26 Puff! The magic dragon, live by the sea…

27 A = 6 9 4 0 3 1 5 7 0 0 ?max(A) ans = 6 9 7 0 3 ?[x,k]=max(A) x =
?max(A) ans = ?[x,k]=max(A) x = k = % index 2019/4/26 Puff! The magic dragon, live by the sea…

28 B = 6 9 4 0 3 ?max(B) ans = 9 Puff! The magic dragon, live by the sea…
?max(B) ans = 9 2019/4/26 Puff! The magic dragon, live by the sea…

29 ?length(A) %傳回m*n (3*2)中的最大值 3
?max(A) ans = ?min(A) ?size(A) ?length(A) %傳回m*n (3*2)中的最大值 3 2019/4/26 Puff! The magic dragon, live by the sea…

30 ?A=[6 2; ;3+4*i 0] A = i ?max(A) ans = ?min(A) 2019/4/26 Puff! The magic dragon, live by the sea…

31 A = i ?sum(A) %將同一行中的所有元素加總 ans = i 2019/4/26 Puff! The magic dragon, live by the sea…

32 ?sort(A) %依同行中元素大小順序排列 ans =
i ?sort(A) %依同行中元素大小順序排列 ans = 2019/4/26 Puff! The magic dragon, live by the sea…

33 2019/4/26 Puff! The magic dragon, live by the sea…

34 2019/4/26 Puff! The magic dragon, live by the sea…

35 2019/4/26 Puff! The magic dragon, live by the sea…

36 Magnitude, length, and absolute value of a Vector
Self testing p.79 2019/4/26 Puff! The magic dragon, live by the sea…

37 多維陣列multidimensional arrays
m*n*q m:列(row), n:行(column), q:頁(page) 第n頁的所有元素 A(:,:,n) 2019/4/26 Puff! The magic dragon, live by the sea…

38 ?A=[4,6,1;5,8,0;3,9,2] A = ?A(:,:,2)=[6,2,9;0,3,1;4,7,5] A(:,:,1) = A(:,:,2) = 2019/4/26 Puff! The magic dragon, live by the sea…

39 ?A=[8 2; 9 5] A = 8 2 9 5 ?B=[4 6; 7 3] B = 4 6 7 3 ?C=cat(3,A,B)
?B=[4 6; 7 3] B = ?C=cat(3,A,B) C(:,:,1) = C(:,:,2) = 2019/4/26 Puff! The magic dragon, live by the sea…

40 cat(1,A,B)=[A;B] 一維 A = ?[A;B] 8 2 ans = 9 5 8 2 B = 9 5 4 6 4 6 7 3
B = ?cat(1,A,B) ans = ?[A;B] ans = 2019/4/26 Puff! The magic dragon, live by the sea…

41 cat(2,A,B)=[A,B] 二維 A = ?[A,B] 8 2 ans = 9 5 B = 8 2 4 6 4 6 9 5 7 3
B = ?cat(2,A,B) ans = ?[A,B] ans = 2019/4/26 Puff! The magic dragon, live by the sea…

42 cat(3,A,B) 三維 A = ans(:,:,2) = 8 2 4 6 9 5 7 3 B = 4 6 7 3 ?C(2,1,1)
B = ?cat(3,A,B) ans(:,:,1) = ans(:,:,2) = ?C(2,1,1) ans = 9 2019/4/26 Puff! The magic dragon, live by the sea…

43 Array Operations B = 4 6 7 3 ?A+B ans = 12 8 16 8 ?A-3*B -4 -16 -12 -4
?A+B ans = ?A-3*B A = ?3*A ans = ?A+1000 ans = ?A/3 2019/4/26 Puff! The magic dragon, live by the sea…

44 Array Operations ?A./B %(B.\A) ans = 2.0000 0.3333 1.2857 1.6667
B = ?A*B ans = ?A.*B ?A./B %(B.\A) ans = ?B./A %(A.\B) ?A\B %(A-1*B) ans = ?A/B %(A*B-1) 2019/4/26 Puff! The magic dragon, live by the sea…

45 2019/4/26 Puff! The magic dragon, live by the sea…

46 2019/4/26 Puff! The magic dragon, live by the sea…

47 2019/4/26 Puff! The magic dragon, live by the sea…

48 Take a rest! Try p. 84 ex.2.3-1 Puff! The magic dragon,
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49 Inner matrix dimensions must agree.
?x'*y ans = ?x*y' 38 ?x=[2,4,-5]; y=[-7,3,-8] ?x.*y ans = ?x*y ??? Error using ==> * Inner matrix dimensions must agree. 2019/4/26 Puff! The magic dragon, live by the sea…

50 p.86, ex.2.3-2 ?v=w-r ?r=[55,36,25]; v = ?w=[-20,59,15]; -75 23 -10
?dist1=sqrt(sum(r.*r)) dist1 = 70.327 ?v=w-r v = ?dist2=sqrt(sum(v.*v)) dist2 = 2019/4/26 Puff! The magic dragon, live by the sea…

51 Ex.2.3-3 ?t=[0:.003:0.5]; ?y=exp(-8*t).*sin(9.7*t+pi/2);
?plot(t,y),xlabel('t (sec)'),... ylabel('y(t)') 2019/4/26 Puff! The magic dragon, live by the sea…

52 p.89, ex.2.3-4 ?d=[560,440,490,530,370] %mile ?t=[10.3, 8.2, 9.1, 10.1, 7.5] % hour ?speed=d./t speed = 2019/4/26 Puff! The magic dragon, live by the sea…

53 Array exponentiation ?A=[4 -5;2 3] ?C=3.^A A = C = 4 -5 81.0000 0.0041
?B=A.^3 B = ?C=3.^A C = 2019/4/26 Puff! The magic dragon, live by the sea…

54 I=V/R, P=V^2/R ?R=[10000, 20000, 35000, , ]; ?V=[120, 80, 110, 200, 350]; ?current=V./R current = ?power=V.^2./R power = 2019/4/26 Puff! The magic dragon, live by the sea…

55 ?L=100*(x/0.6).^(0.625).*((1-x)/0.4).^(-1.625);
?plot(L,x), grid, xlabel('L (mol)'),... ylabel('x (mol B/mol)'), [L, x]=ginput(1) L = x = 0.5219 2019/4/26 Puff! The magic dragon, live by the sea…

56 Multivariable functions
Ex-2.3-7 v=10, 12, 14, 16, 18, 20 m/s =50, 60, 70, 80 degrees g=9.8; v=[10:2:20]; th=[50:10:80] thr=th*(pi/180); vel=[ ]; for k=1:length(th) vel=[vel, v']; end theta=[ ]; for k=1:length(v) theta=[theta; thr]; h=(vel.^2.*(sin(theta)).^2)/(2.*g) H=[v', h]; table=[0, th; H] 2019/4/26 Puff! The magic dragon, live by the sea…

57 2019/4/26 Puff! The magic dragon, live by the sea…

58 Ax=b 2019/4/26 Puff! The magic dragon, live by the sea…

59 2019/4/26 Puff! The magic dragon, live by the sea…

60 2019/4/26 Puff! The magic dragon, live by the sea…

61 2019/4/26 Puff! The magic dragon, live by the sea…

62 Matrix operation Multiplication of vectors p.95
Vector-matrix multiplication p.96 Matrix-matrix multiplication p.96 The general matrix multiplication case p.99 2019/4/26 Puff! The magic dragon, live by the sea…

63 2019/4/26 Puff! The magic dragon, live by the sea…

64 2019/4/26 Puff! The magic dragon, live by the sea…

65 2019/4/26 Puff! The magic dragon, live by the sea…

66 Special matrices Null matrix Identity matrix Puff! The magic dragon,
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67 Special matrices ?eye(3) ?ones(3) ans = ans = 1 0 0 1 1 1 0 1 0 0 0 1
?ones(3) ans = 2019/4/26 Puff! The magic dragon, live by the sea…

68 Special matrices ?zeros(3) ?zeros(2,3) ans = ans = 0 0 0 0 0 0
?zeros(2,3) ans = 2019/4/26 Puff! The magic dragon, live by the sea…

69 Special products cross(A,B) ~ momentum dot(r,F)
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70 2019/4/26 Puff! The magic dragon, live by the sea…

71 2019/4/26 Puff! The magic dragon, live by the sea…

72 polyminals [a1, a2,… an-1, an, an+1] Puff! The magic dragon,
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73 Polynomial algebra f=[9, -5, 3, 7] g=[6, -1, 2] g=[0 g]=[0, 6, -1, 2]
h=f+g=[9, 1, 2, 9] 2019/4/26 Puff! The magic dragon, live by the sea…

74 Polynomial algebra f(x)g(x)=? ?f=[9, -5, 3, 7] ?g=[6, -1, 2]
?product= conv(f,g) product = 2019/4/26 Puff! The magic dragon, live by the sea…

75 Polynomial algebra f(x)/g(x)=? ?f=[9, -5, 3, 7] ?g=[6, -1, 2]
?[quotient, remainder]=deconv(f,g) quotient = %商式 remainder = %餘式 2019/4/26 Puff! The magic dragon, live by the sea…

76 Polynomial algebra f(0), f(2), f(4), f(6), f(8), f(10) f(x)/g(x)=?
?f=polyval(a, x); or f=polyval([9, -5, 3, 7], [0:2:10]) 2019/4/26 Puff! The magic dragon, live by the sea…

77 Self testing P. 108 2019/4/26 Puff! The magic dragon, live by the sea…

78 2019/4/26 Puff! The magic dragon, live by the sea…

79 2019/4/26 Puff! The magic dragon, live by the sea…

80 2019/4/26 Puff! The magic dragon, live by the sea…

81 Summary pp.109 Home work Puff! The magic dragon, live by the sea…
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82 2019/4/26 Puff! The magic dragon, live by the sea…

83 2019/4/26 Puff! The magic dragon, live by the sea…