1. MATLAB Basics With a brief review of linear algebra by Lanyi Xu modified by D.G.E. Robertson

2. 1. Introduction to vectors and matrices MATLAB= MATrix LABoratory What is a Vector? What is a Matrix? Vector and Matrix in Matlab

3. What is a vector A vector is an array of elements, arranged in column, e.g., X is a n-dimensional column vector.

4. In physical world, a vector is normally 3-dimensional in 3-D space or 2- dimensional in a plane (2-D space), e.g.,, or

5. If a vector has only one dimension, it becomes a scalar, e.g.,

6. Vector addition Addition of two vectors is defined by Vector subtraction is defined in a similar manner. In both vector addition and subtraction, x and y must have the same dimensions.

7. Scalar multiplication A vector may be multiplied by a scalar, k, yielding

8. Vector transpose The transpose of a vector is defined, such that, if x is the column vector its transpose is the row vector

9. Inner product of vectors The quantity x T y is referred as the inner product or dot product of x and y and yields a scalar value (or x ∙ y). If x T y = 0 x and y are said to be orthogonal.

10. In addition, x T x, the squared length of the vector x, is The length or norm of vector x is denoted by

11. Outer product of vectors The quantity of xy T is referred as the outer product and yields the matrix

12. Similarly, we can form the matrix xx T as where xx T is called the scatter matrix of vector x.

13. Matrix operations A matrix is an m by n rectangular array of elements in m rows and n columns, and normally designated by a capital letter. The matrix A, consisting of m rows and n columns, is denoted as

14. Where a ij is the element in the i th row and j th column, for i=1,2, ,m and j=1,2,…,n. If m=2 and n=3, A is a 2  3 matrix

15. Note that vector may be thought of as a special case of matrix: a column vector may be thought of as a matrix of m rows and 1 column; a rows vector may be thought of as a matrix of 1 row and n columns; A scalar may be thought of as a matrix of 1 row and 1 column.

16. Matrix addition Matrix addition is defined only when the two matrices to be added are of identical dimensions, i.e., that have the same number of rows and columns. e.g.,

17. For m=3 and n=n:

18. Scalar multiplication The matrix A may be multiplied by a scalar k. Such multiplication is denoted by kA where i.e., when a scalar multiplies a matrix, it multiplies each of the elements of the matrix, e.g.,

19. For 3  2 matrix A,

20. Matrix multiplication The product of two matrices, AB, read A times B, in that order, is defined by the matrix

21. The product AB is defined only when A and B are comfortable, that is, the number of columns is equal to the number of rows in B. Where A is m  p and B is p  n, the product matrix [c ij ] has m rows and n columns, i.e.,

22. For example, if A is a 2  3 matrix and B is a 3  2 matrix, then AB yields a 2  2 matrix, i.e., In general,

23. For example, if and, then

24. and Obviously,.

25. Vector-matrix Product If a vector x and a matrix A are conformable, the product y=Ax is defined such that

26. For example, if A is as before and x is as follow,, then

27. Transpose of a matrix The transpose of a matrix is obtained by interchanging its rows and columns, e.g., if then Or, in general, A=[a ij ], A T =[a ji ].

28. Thus, an m  n matrix has an n  m transpose. For matrices A and B, of appropriate dimension, it can be shown that

29. Inverse of a matrix In considering the inverse of a matrix, we must restrict our discussion to square matrices. If A is a square matrix, its inverse is denoted by A -1 such that where I is an identity matrix.

30. An identity matrix is a square matrix with 1 located in each position of the main diagonal of the matrix and 0s elsewhere, i.e.,

31. It can be shown that

32. MATLAB basic operations MATLAB is based on matrix/vector mathematics Entering matrices Enter an explicit list of elements Load matrices from external data files Generate matrices using built-in functions Create vectors with the colon (:) operator

33. >> x=[1 2 3 4 5]; >> A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1] A = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 >>

36. Generate matrices using built- in functions Functions such as zeros(), ones(), eye(), magic(), etc. >> A=zeros(3) A = 0 0 0 >> B=ones(3,2) B = 1

37. >> I=eye(4)(i.e., identity matrix) I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 >> A=magic(4) (i.e., magic square) A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 >>

38. Generate Vectors with Colon (:) Operator The colon operator uses the following rules to create regularly spaced vectors: j:k is the same as [j,j+1,...,k] j:k is empty if j > k j:i:k is the same as [j,j+i,j+2i,...,k] j:i:k is empty if i > 0 and j > k or if i < 0 and j < k where i, j, and k are all scalars.

39. >> c=0:5 c = 0 1 2 3 4 5 >> b=0:0.2:1 b = 0 0.2000 0.4000 0.6000 0.8000 1.0000 >> d=8:-1:3 d = 8 7 6 5 4 3 >> e=8:2 e = Empty matrix: 1-by-0 Examples

40. Basic Permutation of Matrix in MATLAB sum, transpose, and diag Summation We can use sum() function. Examples, >> X=ones(1,5) X = 1 1 1 1 1 >> sum(X) ans = 5 >>

41. >> A=magic(4) A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 >> sum(A) ans = 34 34 34 34 >>

42. Transpose >> A=magic(4) A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 >> A' ans = 16 5 9 4 2 11 7 14 3 10 6 15 13 8 12 1 >>

43. Expressions of MATLAB Operators Functions

44. Operators +Addition-Subtraction *Multiplication /Division \Left division ^Power ' Complex conjugate transpose ( )Specify evaluation order

45. Functions MATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin. pi3.14159265... iImaginary unit ( ) jSame as i Useful constants:

46. >> rho=(1+sqrt(5))/2 rho = 1.6180 >> a=abs(3+4i) a = 5 >>

47. Basic Plotting Functions plot( ) The plot function has different forms, depending on the input arguments. If y is a vector, plot(y) produces a piecewise linear graph of the elements of y versus the index of the elements of y. If you specify two vectors as arguments, plot(x,y) produces a graph of y versus x.

48. Example, x = 0:pi/100:2*pi; y = sin(x); plot(x,y)

49. Multiple Data Sets in One Graph x = 0:pi/100:2*pi; y = sin(x); y2 = sin(x-.25); y3 = sin(x-.5); plot(x,y,x,y2,x,y3)

50. Distance between a Line and a Point given line defined by points a and b find the perpendicular distance (d) to point c d = norm(cross((b-a),(c-a)))/norm(b-a)