1. Manipulating MATLAB Vector, Matrices 1

2. Variables and Arrays What are variables? You name the variables (as the programmer) and assign them numerical values. You execute the assignment command to place the variable in the workspace memory (memory is part of hardware used for storing information). You are allowed to use the variable in algebraic expressions, etc. once it is assigned. 2

3. An array is MATLAB's basic data structure. Can have any number of dimensions. Most common are: vector - one dimension (a single row or column) matrix - two or more dimensions Arrays can have numbers or letters. (Arrays) 3

4. In MATLAB, a variable is stored as an array of numbers. When appropriate, it is interpreted as a scalar, vector or matrix. The size of an array is specified by the number of rows and the number of columns in the array, with the number of rows indicated first. Variables as Arrays scalar 1 × 1 vector n × 1 or 1 × n matrix n × m 4

5. Scalars are 1×1 arrays. They contain a single value, for example: X=3 Height=5.34 Width=10.09 Scalars 5

6. Vectors A vector is a list of numbers expressed as a 1 dimensional array. A vector can be n×1 or 1×n. Columns are separated by commas (or spaces): x = [1, 2, 3] or x = [1 2 3] Rows are separated by semicolons: y= [1; 2; 3] 6

7. To create a row vector from known numbers, type variable name, then equal sign, then inside square brackets, numbers separated by spaces. variable_name = [ n1, n2, n3 ] >> yr = [1984 1986 1988 1990 1992 1994 1996] yr = 1984 1986 1988 1990 1992 1994 1996 Note MATLAB displays row vector horizontally Commas optional (Creating a row vector) 7

8. To create a column vector from known numbers Method 1 - same as row vector but put semicolon after all but last number variable_name = [ n1; n2; n3 ] >> yr = [1984; 1986; 1988 ] yr = 1984 1986 1988 Note: MATLAB displays column vector vertically (Creating a Column Vector) 8

9. Method 2 - same as row vector but put apostrophe (') after closing bracket. Apostrophe interchanges rows and columns. Will come back on this later. variable_name = [ n1 n2 n3 ]' >> yr = [1984 1986 1988 ]' yr = 1984 1986 1988 (Creating a column vector) 9

10. To create a vector with specified constant spacing between elements variable_name = m:q:n m is the first number n is the last number q is the difference between consecutive numbers v = m:q:n means v = [ m m+q m+2q m+3q... n ] (Constant spacing vectors) 10

11. If q is omitted, the spacing is 1 v = m:n means v = [ m m+1 m+2 m+3... n ] >> x = 1:2:13 x = 1 3 5 7 9 11 13 >> y = 1.5:0.1:2.1 y = 1.5000 1.6000 1.7000 1.8000 1.9000 2.0000 2.1000 Non-integer spacing (Constant spacing vectors) 11

12. >> z = -3:7 z = -3 -2 -1 0 1 2 3 4 5 6 7 >> xa = 21:-3:6 xa = 21 18 15 12 9 6 >> fred = 7:-2:15 fred = Empty matrix: 1-by-0 Negative spacing (Constant spacing vectors) Impossible spacing 12

13. 1. How would you use linspace to set up spacings for planting seedlets in a garden row (30 seedlets, each row 2 meters long). 2. How about planning a 4285km road trip on 21 days. How long each leg would be? (Quick linspace Exercises) 13

14. To create a vector with specified number of terms between first and last v = linspace( xi, xf, n ) xi is first number xf is last number n is the number of terms (obviously must be a positivenumber) (= 100 if omitted) (Fixed length vectors) 14

15. >> va = linspace( 0, 8, 6 ) va = 0 1.6000 3.2000 4.8000 6.4000 8.0000 >> va = linspace( 30, 10, 11 ) va=30 28 26 24 22 20 18 16 14 12 10 m:q:n lets you directly specify spacing. linspace() lets you directly specify number of terms. Six elements Decreasing elements T I P 15

16. Indexing Into an Array allows you to change a value

17. Adding Elements

18. If you add an element outside the range of the original array, intermediate elements are added with a value of zero

19. 2-D Matrices can also be entered by listing each row on a separate line C = [-12, 0, 0 12, 13, 40 1, -1, 90 0, 0, 2] 2-D Matrices

20. M = [12, 152, 6, 197, 32, -32, … 55, 82, 22, 10]; Use an ellipsis to continue a definition onto a new line 2-D Matrices

21. 2-D matrix These semicolons are optional

22. Define a matrix using other matrices as components

23. Or…

24. Matrices A matrix is a two dimensional array of numbers. For example, this is a 4×3 matrix: mat1=[1.0,1.8,1.6; 3.0,- 2.0,25.1; 0.0,-5.1,12.7; 2.3,0.3,-3.1] 123 11.01.81.6 23.0-2.025.1 30.0-5.112.7 42.30.3-3.1 Columns Rows 24

25. Indexed-location of numbers in an array Each item in an array is located in the (row, column). mat1(2,3) ans= 25.1000 123 11.01.81.6 23.0-2.025.1 30.0-5.112.7 42.30.3-3.1 Columns Rows 25

26. Enter the following into MATLAB: Scalar: x=90 Vectors: y=[1, 0, 55] z=[1 0 55] Matrix: m= [ 55, 44, 33; 0, 2, 88] Practice! 26

27. Vector, Matrices 27 Manipulating MATLAB

28. Defining (or assigning) arrays 28 An array can be defined by typing in a list of numbers enclosed in square brackets: Commas or spaces separate numbers. m=[12, 1, -33] or m=[12 1 -33] Semicolons indicate a new row. n=[2, 50, 20;1,1,2;0,-2,66] m= 12 18 -33 12 18 -33 n = 2 50 20 2 50 20 1 1 2 1 1 2 0 -2 66 0 -2 66

29. p = 12 18 -33 12 18 -33 12 18 -33 12 18 -33 2 50 20 2 50 20 2 50 20 2 50 20 1 1 2 1 1 2 1 1 2 1 1 2 0 -2 66 0 -2 66 0 -2 66 0 -2 66 Defining arrays continued 29 You can define an array in terms of another array: r=[m;n] p=[r,r] r = r = 12 18 -33 12 18 -33 2 50 20 2 50 20 1 1 2 1 1 2 0 -2 66 0 -2 66

30. We can access (read from or write to) elements in an array (vector or matrix) individually or in groups. Useful for changing a subset of elements. Useful for making a new variable from a subset of elements. (Verctor/Matrix Addressing) 30

31. (Accessing Vectors Elements 1) 31  You can access any element by indexing the vector name with parentheses (indexing starts from 1, not from 0 as in C). >> odd = [1:3:10] odd = 1 4 7 10 >> odd(3) ans = 7

32. Review… 32 Index – a number used to identify elements in an array Retrieving a value from an array: n(2,1) ans=1 You can change a value in an element in an array with indexing: n(3,2)=55 n = 2 50 20 2 50 20 1 1 2 1 1 2 0 -2 66 0 -2 66 n = 2 50 20 2 50 20 1 1 2 1 1 2 0 55 66 0 55 66 You can extend an array by defining a new element: m=[12 1 -33] m(6)=9 m=[12 1 -33 0 0 9] Notice how undefined values of the array are filled with zeros

33. 33  We can also access a range of elements (a subset of the original vector) by using the colon operator and giving a starting and ending index. >> odd(2:4) ans = 4 7 10  Simple arithmetic between scalars and vectors is possible. >> 10 + [1 2 3] ans = 11 12 13 (Range of Elements / Vector Arithmetic)

34. (Matrices) 34 Create a two-dimensional matrix like this m = [ row 1 numbers; row 2 numbers;... ; last row numbers ] Each row separated by semicolon. All rows have same number of columns >> a=[ 5 35 43; 4 76 81; 21 32 40] a = 5 35 43 4 76 81 21 32 40

35. (Matrices) 35 >> cd=6; e=3; h=4; >> Mat=[e, cd*h, cos(pi/3);... h^2 sqrt(h*h/cd) 14] Mat = 3.0000 24.0000 0.5000 16.0000 1.6330 14.0000 Comma is optional

36. (Matrices) 36 Can also use m:p:n or linspace() to make rows Make sure each row has same number of columns >> A=[1:2:11; 0:5:25;... linspace(10,60,6); 67 2 43 68 4 13] A = 1 3 5 7 9 11 0 5 10 15 20 25 10 20 30 40 50 60 67 2 43 68 4 13

37. (Matrices) 37 What if the number of columns is different? >> B= [ 1:4; linspace(1,4,5) ] ??? Error using ==> vertcat CAT arguments dimensions are not consistent. All arrys in the argument list must have the same number of columns. Four columns Five columns

38. (Special Matrix Commands) 38 zeros(m,n) - makes a matrix of m rows and n columns, all with zeros. ones(m,n) - makes a matrix of m rows and n columns, all with ones. eye(n) - makes a square matrix of n rows and columns. Main diagonal (upper left to lower right) has ones, all other elements are zero.

39. Examples! 39 zeros(2,5) ans = 0 0 0 0 0 ones(3,5) ans = 1 1 1 1 1 eye(5) ans = 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 Note: Placing a single number inside either function will return an n × n array. e.g. ones(5) will return a 5 × 5 array filled with ones.

40. >> zr=zeros(3,4) zr = 0 0 0 0 0 0 0 0 >> ne=ones(4,3) ne = 1 1 1 1 1 1 >> idn=eye(5) idn = 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 40

41. To make a matrix filled with a particular number, multiply ones(m,n) by that number >> z=100*ones(3,4) z = 100 100 100 100 T I P 41

42. (About variables in MATLAB) 42 All variables are arrays Scalar - array with only one element Vector - array with only one row or column Matrix - array with multiple rows and columns Assigning to variable specifies its dimension Don't have to define variable size before assigning to it, as you do in many programming languages Reassigning to variable changes its dimension to that of assignment.

43. (The Transpose Operator) 43 Transpose a variable by putting a single quote after it, e.g., x' In math, transpose usually denoted by superscript "T", e.g., xT Converts a row vector to a column vector and vice-versa. Switches rows and columns of a matrix, i.e., first row of original becomes first column of transposed, second row of original becomes second column of transposed, etc.

44. Continue… 44 The transpose operator, an apostrophe, changes all of an array’s rows to columns and columns to rows. m = [12, 1, -33] m’ m= ans= 12 1 -33 12 1 -33 >> aa=[3 8 1] aa = 3 8 1 >> bb=aa' bb = 3 8 1