1. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Part 3 - Chapter 8 Linear Algebraic Equations and Matrices

2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Review of Matrix Algebra Concepts Let’s take a look at a problem that can be formulated using matrices 2

3. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3 Suppose there are three jumpers connected by bungee cords, so that each cord is fully extended, but unstretched. After they are released, gravity takes hold and the jumpers will eventually come to equilibrium Suppose you are asked to compute the displacement of each of the jumpers – that is x 1, x 2, x 3.

4. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Free body diagrams 4 Assume that each cord behaves like a linear spring, and follows Hooke’s Law

5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Force Balance 5 Since the jumpers are at equilibrium the acceleration is equal to 0

6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3 equations and 3 unknowns 6 Rearrange to give …..

7. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Linear system of equations 7 Which can be solved using matrix algebra

8. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix Algebra Chapter 9 in MATLAB for Engineers Applied ForceF = 200 lbf Pivot Point θ = 60 0 20 feet

9. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In this chapter we’ll learn how to… perform the basic operations of matrix algebra solve simultaneous equations using MATLAB matrix operations

10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The difference between an array and a matrix Most engineers use the two terms interchangeably The only time you need to be concerned about the difference is when you perform matrix algebra calculations

11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Arrays Technically an array is an orderly grouping of information –Arrays can contain numeric information, but they can also contain character data, symbolic data etc.

12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix The technical definition of a matrix is a two- dimensional numeric array used in linear algebra Not even all numeric arrays can precisely be called matrices - only those upon which you intend to perform linear transformations meet the strict definition of a matrix.

13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix Algebra used extensively in engineering applications Matrix algebra is different from the array calculations we have performed thus far

14. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Array Operators A.*Bmultiplies each element in array A times the corresponding element in array B A./B divides each element in array A by the corresponding element in array B A.^B raises each element in array A to the power in the corresponding element of array B

15. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Operators used in Matrix Mathematics Transpose Multiplication Division Exponentiation Left Division

16. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Some Matrix Algebra functions Dot products Cross products Inverse Determinants Covered in Applied Numerical Methods

17. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Transpose In mathematics texts you will often see the transpose indicated with superscript T –A T The MATLAB syntax for the transpose is –A'

18. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The transpose switches the rows and columns

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20. Using the transpose with complex numbers When used with complex numbers, the transpose operator returns the complex conjugate

21. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Dot Products The dot product is sometimes called the scalar product the sum of the results when you multiply two vectors together, element by element.

22. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. * || * * + + Equivalent statements

23. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 9.1 MATLAB for Engineers Calculating the Center of Gravity (Center of Mass) Finding the center of gravity of a structure is important in a number of engineering applications The location of the center of gravity can be calculated by dividing the system up into small components.

24. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In a rectangular coordinate system – are the coordinates of the center of gravity –W is the total mass of the system –x1, x2, and x3 etc are the x coordinates of each system component –y1, y2, and y3 etc are the y coordinates of each system component –z1, z2, and z3 etc are the z coordinates of each system component and –W1, W2, and W3 etc are the masses of each system component

25. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In this example… We’ll find the center of gravity of a small collection of the components used in a complex space vehicle

26. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Vehicle Component Locations and Mass Item x, meters y, metersz metersMass Bolt0.1233.50 gram screw1111.50 gram nut1.50.20.50.79 gram bracket2241.75 gram Formulate the problem using a dot product

27. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Input and Output Input –Location of each component in an x-y-z coordinate system – in meters –Mass of each component, in grams Output –Location of the center of gravity

28. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Hand Example Find the x coordinate of the center of gravity Itemx, metersMass, gram x * m, gram meters Bolt 0.1 x3.50= 0.35 screw 1 x1.50= 1.50 nut 1.5 x0.79= 1.1850 bracket 2 x1.75= 3.5 sum7.54 6.535

29. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We know that… The x coordinate is equal to So… =6.535/7.54 = 0.8667 meters This is a dot product

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31. We could use a plot to evaluate our results This plot was created using the interactive plotting tools

32. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix Multiplication Similar to a dot product

33. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix multiplication results in an array where each element is a dot product. In general, the results are found by taking the dot product of each row in matrix A with each column in Matrix B

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35. Because matrix multiplication is a series of dot products –the number of columns in matrix A must equal the number of rows in matrix B –For an mxn matrix multiplied by an nxp matrix m x n n x p These dimensions must match The resulting matrix will have these dimensions

36. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We could use matrix multiplication to solve the problem in Example 9.1, in a single step

37. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix Powers Raising a matrix to a power is equivalent to multiplying it times itself the requisite number of times –A 2 is the same as A*A –A 3 is the same as A*A*A Raising a matrix to a power requires it to have the name number of rows and columns

38. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix Inverse MATLAB offers two approaches –The matrix inverse function inv(A) –Raising a matrix to the -1 power A -1

39. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A matrix times its inverse is the identity matrix Equivalent approaches to finding the inverse of a matrix

40. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Not all matrices have an inverse Called –Singular –Ill-conditioned matrices Attempting to take the inverse of a singular matrix results in an error statement

41. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Determinants Related to the matrix inverse If the determinant is equal to 0, the matrix does not have an inverse The MATLAB function to find a determinant is –det(A)

42. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We’ll talk about determinants more in the next chapter of Applied Numerical Methods

43. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Cross Products sometimes called vector products the result of a cross product is a vector always at right angles (normal) to the plane defined by the two input vectors –orthogonality

44. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Consider two vectors The cross product is equal to…

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46. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

47. Cross Products are Widely Used Cross products find wide use in statics, dynamics, fluid mechanics and electrical engineering problems

48. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Solutions to Systems of Linear Equations

49. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Using Matrix Nomenclature and AX=B

50. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Symbolically and AX=B

51. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We can solve this problem using the matrix inverse approach This approach is easy to understand, but its not the most efficient computationally

52. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Matrix left division uses Gaussian elimination, which is much more efficient, and less prone to round-off error

53. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 8.2 Applied Numerical Methods Bungee Jumpers 53

54. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Linear System of 3 Equations 54 JumperMass (kg)Spring Constant (N/m) Unstretched Cord Length (m) Top (1)605020 Middle (2)7010020 Bottom (3)805020

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57. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 57 Solution using the matrix inverse strategy Solution using left division

58. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Linear Algebraic Equations find wide use in Engineering 58