Introduction to MATLAB 7 for Engineers - The University of Jordan

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Lecture 4 Computer Applications 0933201 Chapter 22-31The array operations are performed between theelements in corresponding locations in the arrays.For example, the array multiplication operation A.*Bresults in a matrix C that has the same size as A and Band has the elements c ij = a ij b i j . For example, ifA =11 5–9 4B =–7 86 2then >>C = A.*B gives this result:C =11(–7) 5(8)–9(6) 4(2)Z.R.K31 More? See pages 87-88.=–77 40–54 82-32The built-in MATLAB functions such assqrt(x) and exp(x) automatically operate onarray arguments to produce an array resultthe same size as the array argument x.Thus these functions are said to bevectorized functions.For example, in the following session the resulty has the same size as the argument x.>>x = [4, 16, 25];>>y = sqrt(x)y =2 4 5Z.R.KHowever, when multiplying or dividingthese functions, or when raising themto a power, you must use element-byelement(.) operations if the argumentsare arrays.For example, to compute:z = (e y sin x) cos 2 x, you must typez = exp(y).*sin(x).*(cos(x)).^2You will get an error message if the sizeof x is not the same as the size of y.The result z will have the same size asx and y.Array Division (./)The definition of array division is similar tothe definition of array multiplication exceptthat the elements of one array are dividedby the elements of the other array. Botharrays must have the same size. Thesymbol for array right division is (./).For example, ifx = [8, 12, 15] y = [–2, 6, 5]Then>>z = x./yGives z = [8/(–2), 12/6, 15/5] = [–4, 2, 3]Z.R.K2-33 More? See pages 89-90. Z.R.K2-34Also, ifA =24 20–9 4 B =Then>>C = A./BgivesC =24/(–4) 20/5–9/3 4/2=–4 53 2–6 4–3 2Array Exponentiation (.^)MATLAB enables us not only to raisearrays to powers but also to raisescalars and arrays to array powers.To perform exponentiation on anelement-by-element basis, we mustuse the .^ symbol.For example, if x = [3, 5, 8],then typing >>x.^3 produces thearray[3 3 , 5 3 , 8 3 ] = [27, 125, 512].Z.R.K2-35 More? See pages 91-92. Z.R.KZ.R.K. 2008 Page 6 of 102-36
Lecture 4 Computer Applications 0933201 Chapter 2We can raise a scalar to an array power.For example: if p = [2, 4, 5], then typing>>3.^p produces the array [3 2 , 3 4 , 3 5 ] = [9, 81, 243].Remember that (.^) is a single symbol. The dotin 3.^p is not a decimal point associated with thenumber 3. The following operations, with the valueof p given here, are equivalent and give thecorrect answer:>> 3.^p>> 3.0.^p>> 3..^p>> (3).^p>> 3.^[2,4,5]Matrix-Matrix Multiplication (*)In the product of two matrices AB, the number ofcolumns in A must equal the number of rows inB. The row-column multiplications form columnvectors, and these column vectors form the matrixresult. The product AB has the same number ofrows as A and the same number of columns as B.For example:6 –210 34 79 8 (6)(9) + (– 2)(– 5) (6)(8) + (– 2)(12)–5 12 = (10)(9) + (3)(– 5) (10)(8) + (3)(12)(4)(9) + (7)(– 5) (4)(8) + (7)(12)=64 2475 1161 116(2.4–4)2-37Z.R.K37 More? See pages 92-95. Z.R.K382-38Use the operator ( * ) to perform matrixmultiplication in MATLAB. The followingMATLAB session shows how to perform thematrix multiplication shown in (2.4–4).>>A = [6,-2;10,3;4,7];>>B = [9,8;-5,12];>>A*Bans =64 2475 1161 116Matrix multiplication does not have thecommutative property; that is, in general, AB BA.A simple example will demonstrate this fact:AB =whereasBA =6 –210 39 8–12 149 8–12 146 –210 3=78 2054 122134 668 65(2.4–6)(2.4–7)Reversing the order of matrix multiplication is acommon and easily made mistake!!!.=2-39Z.R.KZ.R.K392-40More? See pages 97-104.Special MatricesTwo exceptions to the noncommutative propertyare the null or zero matrix, denoted by 0 andthe identity, or unity, matrix, denoted by I.The null matrix contains all zeros and is notthe same as the empty matrix [ ], which has noelements.These matrices have the following properties:0A = A0 = 0IA = AI = AThe identity matrix is a square matrixwhose diagonal elements are all equal to one,with the remaining elements equal to zero.For example, the 4 × 4 identity matrix isI =1 0 0 00 1 0 00 0 1 00 0 0 1The functions eye(n) and eye(size(A))create an n × n identity matrix and an identitymatrix the same size as the matrix A.2-41Z.R.KZ.R.K412-42 More? See page 105.Z.R.K. 2008 Page 7 of 10
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