Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Engr/Math/Physics 25 Chp2 MATLABArrays: Part-2 Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Learning Goals • Learn to Construct 1D Row and Column Vectors • Create MULTI-Dimensional ARRAYS and MATRICES • Perform Arithmetic Operations on Vectors and Arrays/Matrices • Analyze Polynomial Functions

3. Last Time • Vector/Array Scalar Multiplication by Term-by-Term Scaling >> r = [ 7 11 19]; >> v = 2*r v = 14 22 38 • Vector/Array Addition & Subtraction by Term-by-Term Operations • “Tip-toTail” Geometry

4. Multiplying an Array B by a scalar w produces an Array whose elements are the elements of B multiplied by w. Scalar-Array Multiplication • Using MATLAB >> B = [9,8;-12,14]; >> 3.7*B ans = 33.3000 29.6000 -44.4000 51.8000

5. Multiplication of TWO ARRAYS is not nearly as straightforward as Scalar-Array Mult. MATLAB uses TWO definitions for Array-Array multiplication: ARRAY Multiplication also called element-by-element multiplication MATRIX Multiplication (see also MTH6) DIVISION and EXPONENTIATION must also be CAREFULLY defined when dealing with operations between two arrays. Array-Array Multiplication

6. 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×n elements: Array-Array Operations 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

8. If u and v are column vectors, the result of u.*v is a column vector. The Transpose operation z = (x’).*(y’) yields Array-Array Operations cont • 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.

9. The array operations are performed between the elements in corresponding locations in the arrays. For example, the array multiplication operation A.*B results in an array C that has the same size as A and B and has the elements cij = aij bij . For example, if Array-Array Operations cont • Then C = A.*B Yields

10. The built-in MATLAB functions such as sqrt(x) and exp(x) automatically operate on array arguments to produce an array result of the same size as the array argument x Thus these functions are said to be VECTORIZED Some Examples Array-Array Operations cont >> r = [7 11 19]; >> h = sqrt(r) h = 2.6458 3.3166 4.3589 >> u = [1,2,3]; >> f = exp(u) f = 2.7183 7.3891 20.0855

11. However, when multiplying or dividing these functions, or when raising them to a power, we must use element-by-element dot (.) operations if the arguments are arrays. To Calc: z = (eusinr)•cos2r, enter command Array-Array Operations cont >> z = exp(u).*sin(r).*(cos(r)).^2 z = 1.0150 -0.0001 2.9427 • MATLAB returns an error message if the size of r is not the same as the size of u. The result z will have the same size as r and u.

12. The definition of array division is similar to the definition of array multiplication except that the elements of one array are divided by the elements of the other array. Both arrays must have the same size. The symbol for array right division is ./ Recall r = [ 7 11 19] u = [1,2,3] then z = r./u gives Array-Array DIVISION >> z = r./u z = 7.0000 5.5000 6.3333

13. Consider Array-Array DIVISION cont. • Taking C = A./B yields A = 24 20 -9 4 B = -4 5 3 2 >> A./B ans = -6 4 -3 2

14. MATLAB enables us not only to raise arrays to powers but also to raise scalars and arrays to ARRAY powers. Use the .^ symbol to perform exponentiation on an element-by-element basis if x = [3, 5, 8], then typing x.^3 produces the array [33, 53, 83] = [27, 125, 512] We can also raise a scalar to an array power. For example, if p = [2, 4, 5], then typing 3.^p produces the array [32, 34, 35] = [9, 81, 243]. Array EXPONENTIATION

15. Array to Array Power >> A = [5 6 7; 8 9 8; 7 6 5] A = 5 6 7 8 9 8 7 6 5 >> B = [-4 -3 -2; -1 0 1; 2 3 4] B = -4 -3 -2 -1 0 1 2 3 4 >> C = A.^B C = 0.0016 0.0046 0.0204 0.1250 1.0000 8.0000 49.0000 216.0000 625.0000

16. Multiplication of MATRICES requires meeting the CONFORMABILITY condition The conformability condition for multiplication is that the COLUMN dimensions (k x m) of the LEAD matrix A must be EQUAL to the ROW dimension of the LAG matrix B (m x n) If Matrix-Matrix Multiplication • Then

17. Multiplication of A (k x m) and B (m x n) CONFORMABLE Matrices produces a Product Matrix C with Dimensions (k x n) The elements of C are the sum of the products of like-index Row Elements from A, and Column Elements from B; to whit Matrix-Mult. Mechanics

18. A Vector and Matrix May be Multiplied if they meet the Conformability Critera: (1xm)*(mxp) or (kxm)*(mx1) Given Vector-a, Matrix-B, and aB; Find Dims for all Matrix-Vector Multiplication • Then the Dims: a(1x2), B(2x3), c(1x3)

19. Greek letter sigma (Σ, for sum) is another convenient way of handling several terms or variables – The Definition Summation Notation Digression • For the previous example

20. In General the product of Conformable Matrices A & B when Matrix Mult by Σ-Notation • Then Any Element, cij, of Matrix C for • i = 1 to k (no. Rows) j = 1 to n (no. Cols) e.g.;

21. Matrix Mult Example >> A = [3 1.7 -7; 8.1 -0.31 4.6; -1.2 2.3 0.73; 4 -.32 8; 7.7 9.9 -0.17] A = 3.0000 1.7000 -7.0000 8.1000 -0.3100 4.6000 -1.2000 2.3000 0.7300 4.0000 -0.3200 8.0000 7.7000 9.9000 -0.1700 • A is Then 5x3

22. Matrix Mult Example cont >> B = [0.67 -7.6; 4.4 .11; -7 -13] B = 0.6700 -7.6000 4.4000 0.1100 -7.0000 -13.0000 >> C = A*B C = 58.4900 68.3870 -28.1370 -121.3941 4.2060 -0.1170 -54.7280 -134.4352 49.9090 -55.2210 • B is Then 3x2 • Result, C, is Then 5x2

23. Matrix multiplication is generally not commutative. That is, AB ≠ BAevenif BA is conformable Consider an Illustrative Example Matrix-Mult NOT Commutative

24. Two EXCEPTIONS to the NONcommutative property are the NULL or ZERO matrix, denoted by 0 and 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. Commutation of the Null & Identity Matrices Commutation Exceptions • Strictly speaking 0 & I are always SQUARE

25. Identity Matrix is a square matrix and also it is a diagonal matrix with 1’s along the diagonal similar to scalar “1” Identity and Null Matrices • Null Matrix is one in which all elements are 0 • similar to scalar “0” • Both are “idempotent” Matrices: for A = 0 or I →

26. Use the eye(n) command to Form an nxn Identity Matrix eye and zeros >> I = eye(5) I = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 • Use the zeros(mxn) to Form an mxn 0-Filled Matrix • Strictly Speaking a NULL Matrix is SQUARE >> Z24 = zeros(2,4) Z24 = 0 0 0 0 0 0 0 0

27. Function conv(a,b) computes the product of the two polynomials described by the coefficient arrays a and b. The two polynomials need not be the same degree. The result is the coefficient array of the product polynomial. function [q,r] = deconv(num,den) produces the result of dividing a numerator polynomial, whose coefficient array is num, by a denominator polynomial represented by the coefficient array den. The quotient polynomial is given by the coefficient array q, and the remainder polynomial is given by the coefficient array r. PolyNomial Mult & Div

28. Find the PRODUCT for PolyNomial Mult Example >> f = [2 -7 9 -6]; >> g = [13,-5,3]; >> prod = conv(f,g) prod = 26 -101 158 -144 57 -18

29. Find the QUOTIENT PolyNomial Quotient Example >> f = [2 -7 9 -6]; >> g = [13,-5,3]; >> [quot1,rem1] = deconv(f,g) quot1 = 0.1538 -0.4793 rem1 = 0 0.0000 6.1420 -4.5621

30. The function roots(h) computes the roots of a polynomial specified by the coefficient array h. The result is a column vector that contains the polynomial’s roots. PolyNomial Roots >> r = roots([2, 14, 20]) r = -5 -2 >> rf = roots(f) rf = 2.0000 0.7500 + 0.9682i 0.7500 - 0.9682i >> rg = roots(g) rg = 0.1923 + 0.4402i 0.1923 - 0.4402i

31. The function polyval(a,x)evaluates a polynomial at specified values of its independent variable x, which can be a matrix or a vector. The polynomial’s coefficients of descending powers are stored in the array a. The result is the same size as x. Plot over (−4 ≤ x ≤ 7) the function Plotting PolyNomials Time For Live Demo

32. The Demo Plot

33. All Done for Today • Not Covered in Chapter 2 • §2.6 = Cell Arrays • §2.7 = Structure Arrays Cell-Arrays&StructureArrays

34. Engr/Math/Physics 25 Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

35. Chp2 Demo x = linspace(-4,7,20); p3 = [2 -7 9 -6]; y = polyval(p3,x); plot(x,y, x,y, 'o', 'LineWidth', 2), grid, xlabel('x'),... ylabel('y = f(x)'), title('f(x) = 2x^3 - 7x^2 + 9x - 6')

36. Chp2 Demo >> f = [2 -7 9 -6]; >> x = [-4:0.02:7]; >> fx = polyval(f,x); >> plot(x,fx),xlabel('x'),ylabel('f(x)'), title('chp2 Demo'), grid