1. MATLAB– MATRIX 專論 ME2002-2010Fall (week06- Oct27) 原始檔案 : NTU- 初探 MATLAB2(A++)-for-Week0506.ppt

2. 1. array multiplication (element-by-element multiplication), 2. matrix multiplication. 兩種乘法的定義

3. [1 2] [3 4]

4. 兩純量之間的運算

5. 兩陣列與矩陣之間的運算兩陣列與矩陣之間的運算

6. Symbol + - + -.*./.\.^ Examples [6,3]+2=[8,5] [8,3]-5=[3,-2] [6,5]+[4,8]=[10,13] [6,5]-[4,8]=[2,-3] [3,5].*[4,8]=[12,40] [2,5]./[4,8]=[2/4,5/8] [2,5].\[4,8]=[2\4,5\8] [3,5].^2=[3^2,5^2] 2.^[3,5]=[2^3,2^5] [3,5].^[2,4]=[3^2,5^4] Operation Scalar-array addition Scalar-array subtraction Array addition Array subtraction Array multiplication Array right division Array left division Array exponentiation Form A + b A – b A + B A – B A.*B A./B A.\B A.^B Element-by-element operations

7. Array or Element-by-element multiplication is defined only for arrays having the same size. The definition of the product x.*y, where x and y each have n elements, is x.*y = [x(1)y(1), x(2)y(2),..., x(n)y(n)] if x and y are row vectors. For example, if x = [2, 4, – 5], y = [– 7, 3, – 8] then z = x.*y gives z = [2(– 7), 4 (3), –5(–8)] = [–14, 12, 40] Element-by-element multiplication

8. If x and y are column vectors, the result of x.*y is a column vector. For example z = (x’).*(y’) gives Note that x’ is a column vector with size 3 × 1 and thus does not have the same size as y, whose size is 1 × 3. Thus for the vectors x and y the operations x’.*y and y.*x’ are not defined in MATLAB and will generate an error message. 2(–7) 4(3) –5(–8) –14 12 40 = z = Element-by-element multiplication

9. The array multiplication operation A.*B results in a matrix C that has the same size as A and B and has the elements c i j = a i j b i j. For example, if then C = A.*B gives this result: A = 11 5 –9 4 B = –7 8 6 2 C = 11(–7) 5(8) –9(6) 4(2) = –77 40 –54 8 Element-by-element multiplication

10. The symbol for array right division is./. For example, if x = [8, 12, 15] y = [–2, 6, 5] then z = x./y gives z = [8/(–2), 12/6, 15/5] = [–4, 2, 3] Array Division A = 2420 – 9 4 B = –4 5 3 2 Also, if then C = A./B gives C = 24/(–4) 20/5 –9/3 4/2 = –6 4 –3 2

11. 練習 A=[1 0 ; 2 1] B=[-1 2; 0 1] C=[3; 2] D=5 (a) A+B=? (b) A.* B=? (c) A*B=? (d) A*C=? (e) A+C=? (f) A+D=? (g) A.*D=? (h) A*D=? (a) [0 2;2 2] (b) [-1 0;0 1] (c) [-1 2;-2 5] (d) [3;8] (e) NA (f) [6 5;7 6] (g) [5 0;10 5] (h) [5 0;10 5]

12. 6x + 12y + 4z = 70 7x – 2y + 3z = 5 2x + 8y – 9z = 64 >>A = [6,12,4;7,-2,3;2,8,-9]; >>B = [70;5;64]; >>Solution = A\B Solution = 3 5 -2 The solution is x = 3, y = 5, and z = –2. Solution of Linear Algebraic Equations A X = B X = A -1 B

13. 範例 五條卡車貨運路線的行駛距離與行駛時間如下表：請 問有最高行車速率的路線是哪一條？速率為多少？ 12345 距離 miles 560440490530370 時間 hrs 10.38.29.110.17.5 >>d= [560, 440, 490, 530, 370]; >>t=[10.3,8.2,9.1, 10.1, 7.5]; >>spd = d./ t

14. 範例 電阻 (R) 與電壓 (V) 數據如下表，請其出各自的 電流 (I=V/R) 與消耗的功率 (P=V 2 /R). 12345 R (Ohms) 10 4 2x10 4 3.5x10 4 10 5 2x10 5 V (volts) 12080110200350 R=[ … ]; V=[..]; // fill in the data, not BLNAK. Current=V. / R; Power=V.^2./ R;

15. To perform exponentiation on an element-by-element basis, we must use the.^ symbol. For example, if x = [3, 5, 8], then typing x.^3 produces the array [3 3, 5 3, 8 3 ] = [27, 125, 512]. if p = [2, 4, 5], then typing 3.^p produces the array [3 2, 3 4, 3 5 ] = [9, 81, 243]. Array Exponentiation 3.^p 3.0.^p 3..^p (3).^p 3.^[2,4,5] 結果都一樣 你看得出來嗎？

16. 範例 依據牛頓運動定律，初速 v ，仰角 θ ，物體 的最高高度為 h = v 2 *sin 2 (θ)/(2*g) 請建立一個表格，說明不同 v 與 θ 之下的 h 。 其中 v=10,12,14,16,18,20 m/s θ=50, 60, 70, 80 degree

17. 範例 v= [10:2:20]; th=[50:10:80];thr=th*(pi/180);g=9.8; vel=[];%create the 6x4 array of speeds for k=1:length(thr) vel= [vel,v’]; end theta=[];%create the 6x4 array of angles for k=1:length(v) Theta=[theta;thr]; end h=(vel.^2.*(sin(theta)).^ 2 )/ (2*g); h= [v’,h]; table=[0,th;H)

18. In the product of two matrices AB, the number of columns in A must equal the number of rows in B. The row-column multiplications form column vectors, and these column vectors form the matrix result. The product AB has the same number of rows as A and the same number of columns as B. For example, 6 –2 10 3 4 7 9 8 –5 12 = (6)(9) + (– 2)(– 5) (6)(8) + (– 2)(12) (10)(9) + (3)(– 5) (10)(8) + (3)(12) (4)(9) + (7)(– 5) (4)(8) + (7)(12) 64 24 75 116 1 116 = Matrix Multiplication

19. Use the operator * to perform matrix multiplication in MATLAB. >>A = [6,-2;10,3;4,7]; >>B = [9,8;-5,12]; >> A*B ans = 64 24 75 116 1 116 Matrix Multiplication

20. 範例 有三個產品均需要經過四道工序，所需時間與各工 序的單位時間成本如下表，請問每一產品的生產成 本各是多少？生產 10/5/7 個產品 1/2/3 故需多少成本？ 產出一單位產品需要的時間， hrs 工序 成本 ＄ /hr 產品 1 產品 2 產品 3 車削 10 654 研磨 12 231 銑 14 325 焊接 9 403

21. 範例 A=[10,12,14,9];% hourly cost B=[6 2 3 4; 5 3 2 0; 4 1 5 3]; % time required unitCst=A*B ’ %answer is [162 114 149] C=[10 5 7];%No. of items totalCst=C*unitCst ’ %answer is 3233

22. 範例 U=[6,2,1;2,5,4;4,3,2;9,7,3];% 4 x 3 matrix P=[10,12,13,15;8,7,6,4;12,10,13,9;6,4,11,5];% 4 x 4 matrix C=U ’ *P Quarterly_cst=sum(C)%answer=[400 351 509 355] Category_cst=sum(C ’ )%answer=[760 539 316] 單位成本 $x10 3 產品材料人工運輸 1621 2254 3432 4973 各季產量 產品第1季第1季第2季第2季第3季第3季第4季第4季 110121315 28764 31210139 464115

23. that is, in general, AB  BA. A simple example will demonstrate this fact: AB = 6 –2 10 3 9 8 –12 14 = 78 20 54 122 BA = 9 8 –12 14 6 –2 10 3 = 134 6 68 65 whereas Reversing the order of matrix multiplication is a common and easily made mistake. Matrix multiplication does not have the commutative property

24. the null or zero matrix, denoted by 0 the identity, or unity, matrix, denoted by I. The null matrix contains all zeros and is not the same as the empty matrix [ ], which has no elements. These matrices have the following properties: 0A = A0 = 0 IA = AI = A Two exceptions to the noncommutative property

25. is a square matrix whose diagonal elements are all equal to one, with the remaining elements equal to zero. For example, the 2 × 2 identity matrix is I = 1 0 0 1 The functions eye(n) and eye(size(A)) create an n × n identity matrix and an identity matrix the same size as the matrix A. The identity matrix

26. 建立大小為 m × n 的矩陣 >> A = [1 2 3 4; 5 6 7 8; 9 10 11 12]; % 建立 3×4 的矩陣 A >> A % 顯示矩陣 A 的內容 A = 1 2 3 4 5 6 7 8 9 10 11 12

27. m x n 矩陣的各種處理 % 將矩陣 A 第二列、第三行的元素值，改變為 5 >> A(2,3) = 5 A = 1 2 3 4 5 6 5 8 9 10 11 12 % 取出矩陣 A 的第二橫列、第一至第三直行，並儲存成矩陣 B >> B = A(2,1:3) B = 5 6 5

28. m x n 矩陣的各種處理 ( 續 1) % 將矩陣 B 轉置後、再以行向量併入矩陣 A >> A = [A B'] A = 1 2 3 4 5 5 6 5 8 6 9 10 11 12 5 % 刪除矩陣 A 第二行（：代表所有橫列， [] 代表空矩陣） >> A(:, 2) = [] A = 1 3 4 5 5 5 8 6 9 11 12 5

29. m x n 矩陣的各種處理 ( 續 2) >> A = [A; 4 3 2 1] % 在原矩陣 A 中，加入第四列 A = 1 3 4 5 5 5 8 6 9 11 12 5 4 3 2 1 >> A([1 4], :) = [] % 刪除第 1, 4 列（：代表所有直行， [] 是空矩陣） A = 5 5 8 6 9 11 12 5

30. Colon (:) Command >>clear; clc; >>x=0:1:10; >>A=magic(x(2)); >>B=A(:,2); >>C=A(2,:); >>D=A(2,1:2); >>E=[A B]; >>F=[A;C]; 請寫下 由 A 到 F 的矩陣的內容

31. Array Addressing The colon operator selects individual elements, rows, columns, or ''subarrays'' of arrays. Here are some examples: v(:) represents all the row or column elements of the vector v. v(2:5) represents the second through fifth elements; that is v(2), v(3), v(4), v(5). A(:,3) denotes all the elements in the third column of the matrix A. A(:,2:5) denotes all the elements in the second through fifth columns of A. A(2:3,1:3) denotes all the elements in the second and third rows that are also in the first through third columns.

32. You can use array indices to extract a smaller array from another array. For example, if you first create the array B B = C =C = 16 3 7 8 4 9 2 4 10 13 16 3 7 18 8 4 9 25 3 12 15 17 then type C = B(2:3,1:3), you can produce the following array:

33. m x n 矩陣的各種處理 ( 續 3) array1=[1.1 -2.2 3.3 -4.4 5.5]; array1(3)=? [3.3] array1([1 4])=? [1.1 -4.4] array1(1:2:5)=? [1.1 3.3 5.5] arr2=[1 2 3; -2 -3 -4; 3 4 5]; arr2(1,:)=? [1 2 3] arr2(:,1:2:3)=? [1 3; -2 -4; 3 5]

34. m x n 矩陣的各種處理 5 >>arr3=[1 2 3 4 5 6 7 8]; arr3(5:end)=[5 6 7 8] arr3(end)=[8] >>arr4=[1 2 3 4; 5 6 7 8; 9 10 11 12]; arr4(2:end;2:end)=? [6 7 8; 10 11 12] >>arr4(1:2, [1 4])=[20 21;22 23]; arr4 =? [20 2 3 21; 22 6 7 23; 9 10 11 12] >>arr4=[20 21; 22 23]; arr4=?

35. m x n 矩陣的各種處理 6 >>arr4=[1 2 3 4; 5 6 7 8; 9 10 11 12]; >>arr4(1:2,1:2)=1 arr4= 1 1 3 4 1 1 7 8 9 10 11 12

36. m x n 矩陣的各種處理 7 >>arr5=zeros(4) >>arr6=ones(4) >>arr7=eye(4) % 單元矩陣 比較 length(arr?) 與 size(arr?) 的差別 >>arr5=zeros(3,4) >>arr6=ones(3,4) >>arr7=eye(3,4) 比較 length(arr?) 與 size(arr?) 的差別

37. m x n 矩陣的初始格式化

38. Array Functions size(A)Returns a row vector [m n] containing the sizes of the m x n array A. sort(A)Sorts each column of the array A in ascending order and returns an array the same size as A. sum(A)Sums the elements in each column of the array A and returns a row vector containing the sums. max(A)Returns the algebraically largest element in A if A is a vector. Returns a row vector containing the largest elements in each column if A is a matrix. If any of the elements are complex, max(A) returns the elements that have the largest magnitudes. min(A)Like max but returns minimum values. Length(A) Computes either the number of elements of A if A is a vector or the largest value of m or n if A is an m × n matrix.

39. Advanced Array Functions [b, k] = sort(A)Sorts each column of the array A in ascending order and returns a row vector b and their indices in the row vector k. [x, k] = max(A)Similar to max(A) but stores the maximum values in the row vector x and their indices in the row vector k. [x, k] = min(A)Like max but returns minimum values. [u,v,w] = find(A)Computes the arrays u and v, containing the row and column indices of the nonzero elements of the matrix A, and the array w, containing the values of the nonzero elements. The array w may be omitted.

40. max(A) returns the vector [6,2]; min(A) returns the vector [-10, -5]; size(A) returns [3, 2]; length(A) returns 3. sum(A) returns [-1 -3] 得到 ” 行加總 ” 請問如何求列加總 ？ sum(A’)’ A = 6 2 –10 0 3 – 5 練習

41. 範例 五條卡車貨運路線的行駛距離與行駛時間如下表：請 問有最高行車速率的路線是哪一條？速率為多少？ 12345 距離 miles 560440490530370 時間 hrs 10.38.29.110.17.5 >>d=[560,440,490,530,370]; >>t=[10.3,8.2,9.1,10.1,7.5]; >>[h_spd, route]=max(d./t)

42. 練習 sort 指令 a = [92, 95, 58, 75, 69, 82] ，執行 sort 指令： [b, index] = sort(a) 會得到 – b = [58, 69, 75, 82, 92, 95] – 及 index = [3, 5, 4, 6, 1, 2] – 其中 index 的每個元素代表 b 的每個元素在 a 的位置， – 滿足 b 等於 a(index)