1. 1 Linear Algebraic Equations and Matrices

2. Three Individuals Connected by Bungee Cords

3. Free-body Diagrams  Newton’s second law 3 Rearrange equations [K] {x} = {b}

4. Newton’s Second Law – Equation of Motion  Mass-spring system (similar to bungee jumpers)  Kirchhoff’s current and voltage rules 4 Resistor circuits

5.  Solved a single equation  Now consider more than one variable and more than one equation Linear Algebraic Equations

6. Linear Systems  Linear equations and constant coefficients  a ij and b i are constants 6

7. Mathematical Background  Write system of equations in matrix-vector form 7

8. Solving Systems of Linear Equations  Two ways to solve systems of linear algebraic equations [A]{x}={b}  Left-division x = A\b  Matrix inversion x = inv(A)*b  Matrix inversion only works for square, non- singular systems  Less efficient than left-division 8

9. 向量的乘法、除法  向量的乘法 >>x=[1 2 3]; >>pi*x % 純量乘向量 ans = 3.1416 6.2832 9.4248 >>x=[1 2 3]; >>y=[4 5 6]; >>x.*y % 向量乘向量 ans = 4 10 18 9  向量的除法運算在代數系統 實未定義，但在 MATLAB 中 x./y 表示 x.\y 表示 >>x=[1 2 3]; >>y=[4 5 6]; >>x./y % 向量除向量 ans = 0.2500 0.4000 0.5000

10. Matrix Notations 10 Column 4 Row 3 (second index) (first index)

11. 矩陣的索引與下標 11

12. Scalars, Vectors, Matrices  MATLAB treats variables as “matrices”  Matrix (m  n) - a set of numbers arranged in rows (m) and columns (n)  Scalar: 1  1 matrix  Row Vector: 1  n matrix  Column Vector: m  1 matrix 12

13. Matrix Operations  Transpose ( 轉置 )  In MATLAB, transpose is A  Trace is sum of diagonal elements  In MATLAB, trace(A) 13

14. Matrix Transpose 14

15. Special Matrices  Symmetric matrices 15 a ij = a ji [A] T = [A]

16. Special Matrices  Diagonal matrix 16 [A][I] = [I][A] = [A]  Identity matrix ( 單位矩陣 )

17. Special Matrices  Banded matrix – all elements are zero, with the exception of a band centered on the main diagonal 17 Tridiagonal – three non-zero bands

18. Special Matrices  Lower triangular  Upper triangular 18

19.  Matrix identity [A] = [B] if and only if a ij = b ij for all i and j  Matrix addition and subtraction [C] = [A] + [B]  C ij = A ij + B ij [C] = [A]  [B]  C ij = A ij  B ij  Commutative [A] + [B] = [B] + [A] [A]  [B] =  [B] + [A]  Associative ( [A] + [B] ) + [C] = [A] + ( [B] + [C] ) ( [A] + [B] )  [C] = [A] + ( [B]  [C] ) ( [A]  [B] ) + [C] = [A] + (  [B] + [C] ) Matrix Operations

20. g = 5 Multiplication of Matrix by a Scalar

21. Matrix Multiplication  Visual depiction of how the rows and columns line up in matrix multiplication 21

22. [A]*[B]  [B]*[A] Matrix Multiplication

23.  Matrix multiplication can be performed only if the inner dimensions are equal 23

24. Matrix Multiplication  Interior dimensions have to be equal  For a vector  We will be using square matrices 24

25. Matrix Multiplications  MATLAB performs matrix multiplication automatically A*B = C  Note: no period ‘.’ (not element-by-element operation)  For vectors A*x = b  Associative ( [A] [B] ) [C] = [A] ( [B] [C] )  Distributive [A] ( [B] + [C] ) = [A] [B] + [A] [C] ([A] + [B] ) [C] = [A] [C] + [B] [C]  Not generally commutative [A] [B]  [B] [A] 25

26. Matrix Inverse  Matrix division is undefined  However, there is a matrix inverse for non-singular square matrices [A] -1 [A] = [A] [A] -1 = [I]  Multiplication of a matrix by the inverse is analogous to division 26

27. Augmentation  Whatever you do to left-hand-side, do to the right- hand side (useful when solving system of equations) 27

28. Augmented Matrix

29. >> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6] A = 1 5 3 -4 2 5 6 -1 3 4 -2 5 -1 3 2 6 >> A' ans = 1 2 3 -1 5 5 4 3 3 6 -2 2 -4 -1 5 6 Create a matrix Matrix transpose MATLAB Matrix Manipulations

30.  Matrix concatenation >> x = [5 1 -2 3]; >> y = [2 -1 4 2]; >> z = [3 -2 1 -5]; >> B = [x; y; x; z] B = 5 1 -2 3 2 -1 4 2 5 1 -2 3 3 -2 1 -5 >> C = A + B C = 6 6 1 -1 4 4 10 1 8 5 -4 8 2 1 3 1 >> C = C – B ( = A) C = 1 5 3 -4 2 5 6 -1 3 4 -2 5 -1 3 2 6

31. MATLAB Matrix Manipulations  Matrix multiplication, element-by-element operation 31 A = 1 5 3 -4 2 5 6 -1 3 4 -2 5 -1 3 2 6 B = 5 1 -2 3 2 -1 4 2 5 1 -2 3 3 -2 1 -5 >> A*B ans = 18 7 8 42 47 5 3 39 28 -13 19 -14 29 -14 16 -21 >> A.*B ans = 5 5 -6 -12 4 -5 24 -2 15 4 4 15 -3 -6 2 -30 A*B  A.*B

32. >> D = [2 4 3 1; 3 -5 1 2; 1 -1 3 2] D = 2 4 3 1 3 -5 1 2 1 -1 3 2 >> A*D ??? Error using ==> * Inner matrix dimensions must agree. >> D*A ans = 18 45 26 9 -6 0 -19 10 6 18 -5 24 A = 1 5 3 -4 2 5 6 -1 3 4 -2 5 -1 3 2 6 Inner dimension must agree A*D  D*A MATLAB Matrix Manipulations

33. >> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6] >> format short; AI = inv(A) AI = -0.2324 0.2520 0.1606 -0.2467 0.2428 -0.1397 0.0457 0.1005 -0.1436 0.2063 -0.0862 0.0104 -0.1123 0.0431 0.0326 0.0718 >> A*AI ans = 1.0000 0 0 -0.0000 -0.0000 1.0000 0 0.0000 0.0000 0 1.0000 -0.0000 0.0000 0 0.0000 1.0000 Matrix Inverse MATLAB Matrix Manipulations

34. >> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6] >> I = eye(4) I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 >> Aug = [A I] Aug = 1 5 3 -4 1 0 0 0 2 5 6 -1 0 1 0 0 3 4 -2 5 0 0 1 0 -1 3 2 6 0 0 0 1 >> [n, m] = size(Aug) n = 4 m = 8 Matrix Augmentation MATLAB Matrix Manipulations

35. [K] {x} = {b} JumperMass (kg)Spring constant (N/m)Unstretched cord length (m) Top (1)605020 Middle (2)7010020 Bottom (3)805020 Bungee Jumpers

36. >> k1 = 50; k2 = 100; k3 = 50; >> K=[k1+k2 -k2 0;-k2 k2+k3 -k3;0 -k3 k3] K = 150 -100 0 -100 150 -50 0 -50 50 >> format short >> g = 9.81; mg = [60; 70; 80]*g mg = 588.6000 686.7000 784.8000 >> x=K\mg x = 41.2020 55.9170 71.6130 >> xi = [20; 40; 60]; >> xf = xi + x xf = 61.2020 95.9170 131.6130 >> x = inv(K)*mg x = 41.2020 55.9170 71.6130 k 1 = 50 k 2 = 100 stiffer cord k 3 = 50 Final positions of bungee jumpers

37. Kirchhoff’s Current and Voltage Rules  Example 8.3 37 i 43 i 32 i 54 i 52 i 65 i 12

38. Kirchhoff’s Current and Voltage Rules  Example 8.3 38

39. >> A = [1 1 1 0 0 0 0 -1 0 1 -1 0 0 0 -1 0 0 1 0 0 0 0 1 -1 0 10 -10 0 -15 -5 5 -10 0 -20 0 0]; >> b = [0 0 0 0 0 200]'; >> current = A\b current = 6.1538 -4.6154 -1.5385 -6.1538 -1.5385