1. A string is an array of characters Strings have many uses in MATLAB Display text output Specify formatting for plots Input arguments for some functions Text input from user or data files (What Are Strings?) 1

2. We create a string by typing characters within single quotes ('). Many programming languages use the quotation mark (") for strings. Not MATLAB! Strings can have letters, digits, symbols, spaces. To type single quote in string, use two consecutive single quotes, e.g., make the string of English "Greg's car" by typing '"Greg''s car"' Examples: 'ad ef ', '3%fr2', '{edcba:21!', 'MATLAB' (Creating Strings) 2

3. You can assign string to a variable, just like numbers. >> name = 'Sting' name = Sting >> police = 'New York''s finest' police = New York's finest 3

4. In a string variable Numbers are stored as an array A one-line string is a row vector Number of elements in vector is number of characters in string >> name = 'Howard the Duck'; >> size( name ) ans = 1 15 4

5. Strings are indexed the same way as vectors and matrices Can read by index Can write by index Can delete by index 5

6. Example: >> word = 'dale'; >> word(1) ans = d >> word(1) = 'v' word = vale >> word(end) = [] word = val >> word(end+1:end+3) = 'ley' word = valley 6

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8. MATLAB stores strings with multiple lines as an array. This means each line must have the same number of columns (characters). >> names = [ 'Greg'; 'John' ] names = Greg John >> size( names ) ans = 2 4 8

9. Problem >> names = [ 'Greg'; 'Jon' ]??? Error using ==> vertcat CAT arguments dimensions are not consistent. Must put in extra characters (usually spaces) by hand so that all rows have same number of characters >> names = [ 'Greg'; 'Jon ' ] Greg Jon 3 characters4 characters Extra space 9

10. Making sure each line of text has the same number of characters is a big pain. MATLAB solves problem with char function, which pads each line on the right with enough spaces so that all lines have the same number of characters. >> question=char('Romeo, Romeo,','Wherefore art thou', 'Romeo?') question = Romeo, Wherefore art thou Romeo? >> size (question) ans = 3 18 (String Padding) 10

11. Three lines of text stored in a 3x18 array. MATLAB makes all rows as long as longest row. First and third rows above have enough space characters added on ends to make each row 18 characters long. Romeo, Romeo, Whereforeartthou Romeo? 11

12. Array Operators A.*B multiplies 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

13. Use + to add two arrays or to add a scalar to an array. Use – to subtract one array from another or to subtract a scalar from an array. When using two arrays (vectors or matrices), they must both have the same dimensions (number of rows and number of columns). Vectors must have the same dimensions (rows and columns), not just the same number of elements (1x5 is not the same as 5x1). Addition and Subtraction 13

14. When adding two matrices A and B, MATLAB adds the corresponding elements, i.e., It adds the element in the first row and first column of A to the element in the first row and column of B. It adds the element in the first row and second column of A to the element in the first row and second column of B. Etc... This called elementwise addition. Addition and Subtraction 14

15. When subtracting two arrays A and B, MATLAB performs an elementwise subtraction. In general, an operation between two arrays that works on corresponding elements is called an elementwise operation. Addition and Subtraction 15

16. When adding a scalar to an array, MATLAB adds the scalar to every element of the array. When subtracting a scalar from an array, MATLAB subtracts the scalar from every element of the array. Addition and Subtraction 16

17. There are two ways of multiplying matrices – matrix multiplication and elementwise multiplication. MATRIX MULTIPLICATION The type used in linear algebra. MATLAB denotes this with an asterisk (*). The number of columns in the left matrix must be same as number of rows in the right matrix. Array Multiplication 17

18. Matrix Multiplication 18

19. Matrix Multiplication 19

20. >> A = randi(3,3) A = 3 3 1 3 2 2 1 1 3 >> B=randi(3,3) B = 3 3 1 1 2 2 3 3 3 >> AB = A*B AB = 15 18 12 17 19 13 13 14 12 >> BA = B*A BA = 19 16 12 11 9 11 21 18 18 >> AB == BA ans = 0 0 1 0 0 0 20

21. When performing matrix multiplication on two vectors: They must both be the same size. One must be a row vector and the other a column vector. If the row vector is on the left, the product is a scalar. If the row vector is on the right, the product is a square matrix whose side is the same size as the vectors. Vector Multiplication 21

22. >> h = [ 2 4 6 ] h = 2 4 6 >> v = [ -1 0 1 ]' v = 0 1 >> h * v ans = 4 >> v * h ans = -2 -4 -6 0 0 0 2 4 6 Vector Multiplication Examples 22

23. dot(a,b) computes the inner (scalar) product. a and b must be of the same size. Any combination of vertical or horizontal vectors. Result is always a scalar. EXAMPLE >> h = [ 2 4 6 ] h = 2 4 6 >> v = [ -1 0 1 ]' v = 0 1 >> dot(h,v) ans = 4 >> dot(v,h) ans = 4 Scalar (Dot) Product 23

24. Description C = dot(A,B) returns the scalar product of the vectors A and B. A and B must be vectors of the same length. When A and B are both column vectors, dot(A,B) is the same as A'*B. For multidimensional arrays A and B, dot returns the scalar product along the first non-singleton dimension of A and B. A and B must have the same size. Examples The dot product of two vectors is calculated as shown: a = [1 2 3]; b = [4 5 6]; c = dot(a,b) c = 32 24

25. cross(a,b) computes the cross product of two vectors.. It results in a vector which is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering. More info: http://en.wikipedia.org/wik i/Cross_product EXAMPLE >> a = [ 1 2 0 ] a = 1 2 0 >> b = [3 4 0 ] b = 3 4 0 >> cross(a,b) ans = 0 0 -2 Cross Product 25

26. Description C = cross(A,B) returns the cross product of the vectors A and B. That is, C = A x B. A and B must be 3-element vectors. Examples The cross and dot products of two vectors are calculated as shown: a = [1 2 3]; b = [4 5 6]; c = cross(a,b) c = -3 6 -3 d = dot(a,b) d = 32 26

27. A square matrix with ones on main diagonal and zeros elsewhere. When we do a matrix multiplication on any array or vector with the identity matrix, the array or vector is unchanged. True whether multiply with identity matrix is on the left or on right MATLAB command eye(n) makes an n×n identity matrix Identity Matrix 27

28. Determinants A determinant is a function associated with square matrices In math, determinant of A is written as det(A) or |A| In MATLAB, compute determinant of A with det(A) A matrix has an inverse only if it is square and its determinant is not zero. 28

29. Determinants Example with Cross Product 29

30. In math, inverse of a matrix A is written as A -1 In MATLAB, get inverse with A^-1 or inv(A) 30

31. Left division, \: Left division is one of MATLAB's two kinds of array division Used to solve the matrix equation AX=B A is a square matrix, X, B are column vectors Solution is X = A -1 B In MATLAB, solve by using left division operator (\), i.e., >> X = A \ B 31

32. When solving set of linear equations, use left division, not inverse, i.e., use X=A\B not X=inv(A)*B Left division is 2-3 times faster Often produces smaller error than inv() Sometimes inv() can produce erroneous results 32

33. Right division, /: Right division is the other kind of MATLAB's array division Used to solve the matrix equation XC=D C is a square matrix, X, D are row vectors Solution is X = D·C -1 In MATLAB, solve by using right division operator (/), i.e., >> X = D / C 33

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35. Array Division 35

36. Another way of saying elementwise operations is element-by-element operations. Addition and subtraction of arrays is always elementwise. Multiplication, division, exponentiation of arrays can be elementwise. Both arrays must be same dimension. Element by Element Operations 36

37. Do elementwise multiplication, division, exponentiation by putting a period in front of the arithmetic operator. Element by Element Operations 37

38. Element by Element Operations 38

39. Page 73 of text book! 39

40. ELEMENTWISE MULTIPLICATION Use.* to get elementwise multiplication (notice period before asterisk) Both matrices must have the same dimensions >> A = [1 2; 3 4]; >> B = [0 1/2; 1 -1/2]; >> C = A.* B >> C = 0 1 3 -2 40

41. If matrices not same dimension in elementwise multiplication, MATLAB gives an error: >> A = [ 1 2; 3 4]; >> B = [1 0]'; >> A.* B % Meant matrix multiplication! ??? Error using ==> times Matrix dimensions must agree. >> A * B % this works ans = 1 3 41

42. Be careful – when multiplying square matrices: Both types of multiplication always work. If you specify the wrong operator, MATLAB will do the wrong computation and there will be no error! Difficult to find this kind of mistake. Element by Element Operations 42

43. EXAMPLE >> A = [1 2; 3 4]; >> B = [0 1/2; 1 -1/2]; >> A.* B >> ans 0 1 3 -2 >> A * B ans = 2.0000 -0.5000 4.0000 -0.5000 43

44. Built-in MATLAB functions can accept arrays as inputs When input is array, output is array of same size with each element being result of function applied to corresponding input element Example: if x is a 7-element row vector, cos(x) is [cos(x1) cos(x2) cos(x3) cos(x4) cos(x5) cos(x6) cos(x7)] Using Arrays in Matlab Built-in Functions 44

45. BUILT-IN FUNCTIONS FOR ANALYZING ARRAYS MATLAB has lots of functions for operating on arrays. For a vector v mean(v) – mean (average) max(v) – maximum value, optionally with index of maximum ([C,I] = max(v)) min(v) – minimum value, optionally with index of minimum ([C,I] = min(v)) sum(v) – sum sort(v) – elements sorted into ascending order 45

46. BUILT-IN FUNCTIONS FOR ANALYZING ARRAYS median(v) – median std(v) – standard deviation dot(v,w) – dot (inner product); v, w both vectors of same size but any dimension cross(v,w) – cross product; v, w must both have three elements but any dimension det(A) – determinant of square matrix A inv(A) – inverse of square matrix A See Table 3-1 in text book for details on the preceding functions. 46

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