ENGR 1181 MATLAB 4: Array Operations

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1 ENGR 81 MTL 4: rray Operations Learning Objectives 1. Explain meaning of element-by-element operations. Identify situations where the standard operators in MTL (when used with arrays) are reserved for linear algebra, which is not always element-by-element 3. pply dot operators for a. The six cases where linear algebra is not element-by-element and therefore dot operators are needed to produce element-by-element calculations Topics Students will read Chapter of the MTL book before coming to class. This preparation material is provided to supplement this reading. Students will learn a basic understanding of array operations in MTL. This material contains the following: 1. Matrix rithmetic Overview a. Matrix Math (beyond this class) b. Element-by-Element rithmetic. Scalar-rray rithmetic 3. rray-rray rithmetic 4. Dot Operators 5. Element-by-Element Examples 6. uilt-in rray Functions 7. pplication of rray Operations 1. Matrix rithmetic Overview There are two types of matrix arithmetic: 1. Matrix math is used in Linear lgebra to solve simultaneous equations. This is beyond the scope of this course, but can be read about in Chapter. 3. and This course introduces element-by-element arithmetic. If = [a b c] and = [x y z], then + = [a+x b+y c+z] 1

2 ENGR 81 MTL 4: rray Operations If and are matrices and s is a scalar, then: There are only six cases where matrix math and element-by-element arithmetic differ * / ^ ^s s^ s/ For these cases, the standard operator has been chosen to represent matrix arithmetic, so we need a new symbol for element-by element arithmetic (dot operators).*./.^.^s s.^ s./ The arrays must be the same size for matrix arithmetic. Element-by-element operations for row vectors: If: a = [a1 a a3] and b = [b1 b b3] Then: a.* b = [ a 1 * b 1 a * b a 3* b 3 ] a./ b = [ a 1 / b 1 a / b a 3 / b 3 ] a.^ b = [ a 1^b 1 a ^b a 3^b 3 ] a.^ = [ a 1^ a ^ a 3^ ]./a = [ /a 1 /a /a 3 ]. Dot Operators Given the matrices and : Then, the following dot operations are true: Multiplication.* Division (vector by vector) /./ 1/ / 1 / 1 3 / / 1 3 / 3 / 3 /

3 ENGR 81 MTL 4: rray Operations Division (scalar by vector) 6./ 6/ 6/ 6/ 1 6/ 6/ 6/ 1 3 6/ 6/ 6/ 3 Exponential (vector to scalar).^ If you forget a dot when one is required, you will either get a cryptic error message (involving matrix dimensions) or you will get the wrong answer. For certain operations, matrix math is the same as element-by-element arithmetic, so dot operators are optional. This is true for the following operations: s+ s- -s + - s* /s 3. Scalar-rray rithmetic Scalar-rray ddition and Subtraction The scalar operates on each element in the array. For example, given the following matrix M and the scalar s = we would get the following results for addition and subtraction. 3

4 ENGR 81 MTL 4: rray Operations Scalar-rray Multiplication and Division The scalar operates on each element in the array For example, given the same matrix M and scalar s = we would get the following results for multiplication and division 4

5 ENGR 81 MTL 4: rray Operations 4. rray-rray rithmetic rray-rray ddition and Subtraction The arrays being used in the operation must have the same size. The sum (difference) of two arrays is obtained by adding (subtracting) corresponding array elements, resulting in a new array of the same size. 5

6 ENGR 81 MTL 4: rray Operations 5. Element-by-Element Examples Given the following matrices and : >> = [1,, 3; 4, 5, 6] = >> = [1,, -1;, -, 5] = then the following operations are true: >>.* >>./ >>.^

7 ENGR 81 MTL 4: rray Operations 6. uilt-in rray Functions MTL has many built-in functions that can be used with arrays. Some of them are ( is a vector): max() min() mean() sum() length() sort() Returns the largest element in Returns the smallest element in Returns the average value of the elements in Returns the sum of the elements of Returns the number of elements in Sorts the elements of The Help window gives information about many other built-in MTL functions. EXMPLES OF UILT IN FUNCTIONS: >> = [ ] =

8 ENGR 81 MTL 4: rray Operations >> max() 14 >> min() >> mean() 8 >> sum() 48 >> length() 6 >> sort()

9 ENGR 81 MTL 4: rray Operations >> mass = [ ]; >> force = [ ]; >> mu = force./ (9.81 * mass) mu = >> mu_ave = mean(mu) meu_ave =

Mathematical Operations with Arrays and Matrices

Mathematical Operations with Arrays and Matrices Array Operators (element-by-element) (important) + Addition A+B adds B and A - Subtraction A-B subtracts B from A.* Element-wise multiplication.^ Element-wise