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CPSC 125 Ch 4 Sec 6
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CPSC 125 Ch 4 Sec 6

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  • 1. Relations, Functions, and Matrices Mathematical Structures for Computer Science Chapter 4 Section 4.6 © 2006 W.H. Freeman & Co. Copyright MSCS Slides Matrices Relations, Functions and Matrices Monday, March 29, 2010
  • 2. Matrix ● Data about many kinds of problems can often be represented using a rectangular arrangement of values; such an arrangement is called a matrix. ● A is a matrix with two rows and three columns. ● The dimensions of the matrix are the number of rows and columns; here A is a 2 × 3 matrix. ● Elements of a matrix A are denoted by aij, where i is the row number of the element in the matrix and j is the column number. ● In the example matrix A, a23 = 8 because 8 is the element in row 2, column 3, of A. Section 4.6 Matrices 2 Monday, March 29, 2010
  • 3. Example: Matrix ● The constraints of many problems are represented by the system of linear equations, e.g.: x + y = 70 24x + 14y = 1180 The solution is x = 20, y = 50 (you can easily check that this is a solution). ● The matrix A is the matrix of coefficients for this system of linear equations. Section 4.6 Matrices 3 Monday, March 29, 2010
  • 4. Matrix ● If X = Y, then x = 3, y = 6, z = 2, and w = 0. ● We will often be interested in square matrices, in which the number of rows equals the number of columns. ● If A is an n × n square matrix, then the elements a11, a22, ... , ann form the main diagonal of the matrix. ● If the corresponding elements match when we think of folding the matrix along the main diagonal, then the matrix is symmetric about the main diagonal. ● In a symmetric matrix, aij = aji. Section 4.6 Matrices 4 Monday, March 29, 2010
  • 5. Matrix Operations ● Scalar multiplication calls for multiplying each entry of a matrix by a fixed single number called a scalar. The result is a matrix with the same dimensions as the original matrix. ● The result of multiplying matrix A: by the scalar r = 3 is: Section 4.6 Matrices 5 Monday, March 29, 2010
  • 6. Matrix Operations ● Addition of two matrices A and B is defined only when A and B have the same dimensions; then it is simply a matter of adding the corresponding elements. ● Formally, if A and B are both n × m matrices, then C = A + B is an n × m matrix with entries cij = aij + bij: Section 4.6 Matrices 6 Monday, March 29, 2010
  • 7. Matrix Operations ● Subtraction of matrices is defined by A − B = A + (− l)B ● In a zero matrix, all entries are 0. If we add an n × m zero matrix, denoted by 0, to any n × m matrix A, the result is matrix A. We can symbolize this by the matrix equation: 0+A=A ● If A and B are n × m matrices and r and s are scalars, the following matrix equations are true: A+B=B+A (A + B) + C = A + (B + C) r(A + B) = rA + rB (r + s)A = rA + sA r(sA) = (rs)A ! ! ! ! rA = Ar Section 4.6 Matrices 7 Monday, March 29, 2010
  • 8. Matrix Operations ● Matrix multiplication is computed as A times B and denoted as A ⋅ B. ● Condition required for matrix multiplication: the number of columns in A must equal the number of rows in B. Thus we can compute A ⋅ B if A is an n × m matrix and B is an m × p matrix. The result is an n × p matrix. ● An entry in row i, column j of A ⋅ B is obtained by multiplying elements in row i of A by the corresponding elements in column j of B and adding the results. Formally, A ⋅ B = C, where Section 4.6 Matrices 8 Monday, March 29, 2010
  • 9. Example: Matrix Multiplication ● To find A ⋅ B = C for the following matrices: ● Similarly, doing the same for the other row, C is: Section 4.6 Matrices 9 Monday, March 29, 2010
  • 10. Matrix Multiplication ● Compute A ⋅ B and B ⋅ A for the following matrices: ● Note that even if A and B have dimensions so that both A ⋅ B and B ⋅ A are defined, A ⋅ B need not equal B ⋅ A. Section 4.6 Matrices 10 Monday, March 29, 2010
  • 11. Matrix Multiplication ● Where A, B, and C are matrices of appropriate dimensions and r and s are scalars, the following matrix equations are true (the notation A (B ⋅ C) is shorthand for A ⋅ (B ⋅ C)): A (B ⋅ C) = (A ⋅ B) C A (B + C) = A ⋅ B + A ⋅ C (A + B) C = A ⋅ C + B ⋅ C rA ⋅ sB = (rs)(A ⋅ B) ● The n × n matrix with 1s along the main diagonal and 0s elsewhere is called the identity matrix, denoted by I. If we multiply I times any nn matrix A, we get A as the result. The equation is: I⋅A=A⋅I=A Section 4.6 Matrices 11 Monday, March 29, 2010
  • 12. Matrix Multiplication Algorithm ALGORITHM MatrixMultiplication //computes n × p matrix A ⋅ B for n × m matrix A, m × p matrix B //stores result in C for i = 1 to n do for j = 1 to p do C[i, j] = 0 for k =1 to m do C[i, j] = C[i, j] + A[i, k] * B[k, j] end for end for end for write out product matrix C ● If A and B are both n × n matrices, then there are Θ(n3) multiplications and Θ(n3) additions required. Overall complexity is Θ(n3) Section 4.6 Matrices 12 Monday, March 29, 2010