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Matlab Training Session 2: Matrix Operations and Relational Operators

Matlab Training Session 2: Matrix Operations and Relational Operators. Course Outline Weeks: Introduction to Matlab and its Interface (Jan 13 2009) Fundamentals (Operators) Fundamentals (Flow) Importing Data Functions and M-Files Plotting (2D and 3D) Statistical Tools in Matlab

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Matlab Training Session 2: Matrix Operations and Relational Operators

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  1. Matlab Training Session 2:Matrix Operations and Relational Operators

  2. Course Outline Weeks: • Introduction to Matlab and its Interface (Jan 13 2009) • Fundamentals (Operators) • Fundamentals (Flow) • Importing Data • Functions and M-Files • Plotting (2D and 3D) • Statistical Tools in Matlab • Analysis and Data Structures Course Website: http://www.queensu.ca/neurosci/matlab.php

  3. Week 2 Lecture Outline Fundamentals of Matlab 1 • Week 1 Review B. Matrix Operations: • The empty matrix • Creating multi-dimensional matrices • Manipulating Matrices • Matrix Operations C. Operators • Relational Operators

  4. Week 1 Review Working with Matrices c = 5.66 or c = [5.66] c is a scalar or a 1 x 1 matrix  x = [ 3.5, 33.22, 24.5 ] x is a row vector or a 1 x 3 matrix  x1 = [ 2 5 3 -1] x1 is column vector or a 4 x 1 matrix A = [ 1 2 4 2 -2 2 0 3 5 5 4 9 ]A is a 4 x 3 matrix

  5. Week 1 Review • Indexing Matrices • A m x n matrix is defined by the number of m rows and number of n columns • An individual element of a matrix can be specified with the notation A(i,j) or Ai,j for the generalized element, or by A(4,1)=5 for a specific element. • Example: • >> A = [1 2 4 5;6 3 8 2] A is a 2 x 4 matrix • >> A(2,1) • Ans 6

  6. Week 1 Review • Indexing Matrices • Specific elements of any matrix can be overwritten using the matrix index • Example: • A = [1 2 4 5 • 6 3 8 2] • >> A(2,1) = 9 • Ans • A = [1 2 4 5 • 9 3 8 2]

  7. Week 1 Review • Indexing Matrices • A = [1 2 4 5 • 6 3 8 2] • The colon operator can be used to index a range of elements • >> A(1,1:3) • Ans 1 2 4

  8. Matrix Indexing Cont.. • Indexing Matrices • A = [1 2 4 5 • 6 3 8 2] • The colon operator can index all rows or columns without setting an explicit range • >> A(:,3) • Ans 4 8 • >> A(2,:) • Ans 6 3 8 2

  9. B. Matrix Operations

  10. Matrix Operations • Indexing Matrices • An empty or null matrix can be created using square brackets • >> A = [ ] • ** TIP: The size and length functions can quickly return the number of elements and dimensions of a matrix variable

  11. Matrix Operations • Indexing Matrices • A = [1 2 4 5 • 6 3 8 2] • The colon operator can can be used to remove entire rows or columns • >> A(:,3) = [ ] • A = [1 2 5 • 6 3 2] • >> A(2,:) = [ ] • A = [1 2 5]

  12. Matrix Operations • Indexing Matrices • A = [1 2 4 5 • 6 3 8 2] • However individual elements within a matrix cannot be assigned an empty value • >> A(1,3) = [ ] • ??? Subscripted assignment dimension mismatch.

  13. N – Dimensional Matrices • A = [1 2 4 5 B = [5 3 7 9 • 6 3 8 2] 1 9 9 8] • Multidimensional matrices can be created by concatenating 2-D matrices together • The cat function concatenates matrices of compatible dimensions together: • Usage: cat(dimensions, Matrix1, Matrix2)

  14. N – Dimensional Matrices • Examples • A = [1 2 4 5 B = [5 3 7 9 • 6 3 8 2] 1 9 9 8] • >> C = cat(3,[1,2,4,5;6,3,8,2],[5,3,7,9;1,9,9,8]) • >> C = cat(3,A,B)

  15. Matrix Operations • Scalar Operations • Scalar (single value) calculations can be can performed on matrices and arrays • Basic Calculation Operators • + Addition • - Subtraction • * Multiplication • / Division • ^ Exponentiation

  16. Matrix Operations • Scalar Operations • Scalar (single value) calculations can be performed on matrices and arrays • A = [1 2 4 5 B = [1 C = 5 • 6 3 8 2] 7 • 3 • 3] • Try: • A + 10; A * 5; B / 2; A.^C; A*B

  17. Matrix Operations • Scalar Operations • Scalar (single value) calculations can be performed on matrices and arrays • A = [1 2 4 5 B = [1 C = 5 • 6 3 8 2] 7 • 3 • 3] • Try: • A + 10 • A * 5 • B / 2 • A^C What is happening here?

  18. Matrix Operations • Matrix Operations • Matrix to matrix calculations can be performed on matrices and arrays • Addition and Subtraction • Matrix dimensions must be the same or the added/subtracted value must be scalar • A = [1 2 4 5 B = [1 C = 5 D = [2 4 6 8 • 6 3 8 2] 7 1 3 5 7] • 3 • 3] • Try: • >>A + B >>A + C >>A + D

  19. Matrix Operations • Matrix Multiplication • Built in matrix multiplication in Matlab is either: • Algebraic dot product • Element by element multiplication

  20. Matrix Operations • The Dot Product • The dot product for two matrices A and B is defined whenever the number of columns of A are equal to the number of rows of b • A(x1,y1) * B(x2,y2)

  21. Matrix Operations • The Dot Product • The dot product for two matrices A and B is defined whenever the number of columns of A are equal to the number of rows of b • A(x1,y1) * B(x2,y2)

  22. Matrix Operations • The Dot Product • The dot product for two matrices A and B is defined whenever the number of columns of A are equal to the number of rows of b • A(x1,y1) * B(x2,y2)

  23. Matrix Operations • The Dot Product • The dot product for two matrices A and B is defined whenever the number of columns of A are equal to the number of rows of b • A(x1,y1) * B(x2,y2) = C(x1,y2)

  24. Matrix Operations • The Dot Product • A(x1,y1) * B(x2,y2) = C(x1,y2) • A = [1 2 B = [1 D = [2 2 E = [2 4 3 6] • 6 3] 7 2 2] • 3 • 3] • Try: • >>A * D • >>B * E • >>A * B

  25. Matrix Operations • Element by Element Multiplication • Element by element multiplication of matrices is performed with the .* operator • Matrices must have identical dimensions • A = [1 2 B = [1 D = [2 2 E = [2 4 3 6] • 6 3 ] 7 2 2 ] • 3 • 3] • >>A .* D • Ans = [ 2 4 • 12 6]

  26. Matrix Operations • Matrix Division • Built in matrix division in Matlab is either: • Left or right matrix division • Element by element division

  27. Matrix Operations • Left and Right Division • Left and Right division utilizes the / and \ operators • Left (\) division: • X = A\B is a solution to A*X = B • Right (/) division: • X = B/A is a solution to X*A = B • Left division requires A and B have the same number of rows • Right division requires A and B have the same number of columns

  28. Matrix Operations • Element by Element Division • Element by element division of matrices is performed with the ./ operator • Matrices must have identical dimensions • A = [1 2 4 5 B = [1 D = [2 2 2 2 E = [2 4 3 6] • 6 3 8 2] 7 2 2 2 2] • 3 • 3] • >>A ./ D • Ans = [ 0.5000 1.0000 2.0000 2.5000 • 3.0000 1.5000 4.0000 1.0000 ]

  29. Matrix Operations • Element by Element Division • Any division by zero will be returned as a NAN in matlab (not a number) • Any subsequent operation with a NAN value will return NAN

  30. Matrix Operations • Matrix Exponents • Built in matrix Exponentiation in Matlab is either: • A series of Algebraic dot products • Element by element exponentiation • Examples: • A^2 = A * A (Matrix must be square) • A.^2 = A .* A

  31. Matrix Operations • Shortcut: Transposing Matrices • The transpose of a matrix is the matrix formed by interchanging the rows and columns of a given matrix • A = [1 2 4 5 B = [1 • 6 3 8 2] 7 • 3 • 3] • >> transpose(A) >> B’ • A = [1 6 B = [1 7 3 3] • 2 3 • 4 8 • 5 2]

  32. Matrix Operations Other handy built in matrix functions Include: inv() Matrix inverse det() Matrix determinant poly() Characteristic Polynomial kron() Kronecker tensor product

  33. C. Relational Operators

  34. Relational Operators • Relational operators are used to compare two scaler values or matrices of equal dimensions • Relational Operators • < less than • <= less than or equal to > Greater than >= Greater than or equal to == equal ~= not equal

  35. Relational Operators • Comparison occurs between pairs of corresponding elements • A 1 or 0 is returned for each comparison indicating TRUE or FALSE • Matrix dimensions must be equal! • >> 5 == 5 • Ans 1 • >> 20 >= 15 • Ans 1

  36. Relational Operators A = [1 2 4 5 B = 7 C = [2 2 2 2 6 3 8 2] 2 2 2 2] Try: >>A > B >> A < C

  37. Relational Operators • The Find Function • The ‘find’ function is extremely helpful with relational operators for finding all matrix elements that fit some criteria • A = [1 2 4 5 B = 7 C = [2 2 2 2 D = [0 2 0 5 0 2] • 6 3 8 2] 2 2 2 2] • The positions of all elements that fit the given criteria are returned • >> find(D > 0) • The resultant positions can be used as indexes to change these elements • >> D(find(D>0)) = 10 D = [10 2 10 5 10 2]

  38. Relational Operators • The Find Function • A = [1 2 4 5 B = 7 C = [2 2 2 2 D = [0 2 0 5 0 2] • 6 3 8 2] 2 2 2 2] • The ‘find’ function can also return the row and column indexes of of matching elements by specifying row and column arguments • >> [x,y] = find(A == 5) • The matching elements will be indexed by (x1,y1), (x2,y2), … • >> A(x,y) = 10 • A = [ 1 2 4 10 • 6 3 8 2 ]

  39. Getting Help • Help and Documentation • Digital • Accessible Help from the MatlabStart Menu • Updated online help from the MatlabMathworks website: • http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.html • Matlab command prompt function lookup • Built in Demo’s • Websites • Hard Copy • Books, Guides, Reference • The Student Edition of Matlab pub. Mathworks Inc.

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