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Matlab : Matrix operations and relational operators (Part 2)

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    Matlab : Matrix operations and relational operators (Part 2) Matlab : Matrix operations and relational operators (Part 2) Presentation Transcript

    • Matlab Training Session 2: Matrix Operations and Relational Operators
        • Course Outline
        • Introduction to Matlab and its Interface
        • Fundamentals (Operators)
        • Fundamentals (Flow)
        • Importing Data
        • Functions and M-Files
        • Plotting (2D and 3D)
        • Statistical Tools in Matlab
        • Analysis and Data Structures
    • Review Working with Matrices c = 5.66 or c = [5.66] c is a scalar or a (1 × 1) matrix  x = [ 3.5, 33.22, 24.5 ] x is a row vector or a (1 × 3) matrix  x1 = [ 2 5 3 -1] x1 is column vector or a (4 × 1) matrix A = [ 1 2 4 2 -2 2 0 3 5 5 4 9 ] A is a (4 × 3) matrix
    • Review
      • Indexing Matrices
      • A (m × 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 × 4) matrix
      • >> A(2,1)
      • Ans 6
    • 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]
    • 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
    • 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
    • B. Matrix Operations
    • 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
    • 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]
    • 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.
    • 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)
    • 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)
    • N – Dimensional Matrices
    • Matrix Operations
      • Scalar Operations
      • Scalar (single value) calculations can be performed on matrices and arrays
        • Basic Calculation Operators
        • + Addition
        • - Subtraction
        • * Multiplication
        • / Division
        • ^ Exponentiation
    • 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
    • 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
    • Matrix Operations
      • Matrix Multiplication
      • Built in matrix multiplication in Matlab is either:
      • A lgebraic dot product
      • Element by element multiplication
    • 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
    • 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]
    • Matrix Operations
      • Matrix Division
      • Built in matrix division in Matlab is either:
      • Left or right matrix division
      • Element by element division
    • Matrix Operations
      • Left and Right Division
      • Left and Right division utilizes the / and operators
      • Left () division:
      • X = AB 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
    • 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 ]
    • 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
    • Matrix Operations
      • Matrix Exponents
      • Built in matrix Exponentiation in Matlab is either:
      • A series of A lgebraic dot products
      • Element by element exponentiation
      • Examples:
      • A^2 = A * A (Matrix must be square)
      • A.^2 = A .* A
    • Matrix Operations
      • Shortcut: Transposing Matrices
      • The transpose of a matrix is the matrix formed by interchanging the rows and columns of a given matrix
      >> B’ B=[1 7 3 3] >> transpose(A) A=[1 6 2 3 4 8 5 2] B = [1 7 3 3] A = [1 2 4 5 6 3 8 2]
    • Matrix Operations Other handy built in matrix functions Include: inv() Matrix inverse det() Matrix determinant poly() Characteristic Polynomial kron() Kronecker tensor product
    • C. Relational Operators
    • 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
    • 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
    • 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
    • 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]
    • 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 ]