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Lecture 6: Matrices

Lecture 6: Matrices. Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about matrices and their relationship with relations Become familiar with Boolean matrices Learn the relationship between Boolean matrices and different closures of a relation

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Lecture 6: Matrices

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  1. Lecture 6: Matrices Discrete Mathematical Structures: Theory and Applications

  2. Learning Objectives • Learn about matrices and their relationship with relations • Become familiar with Boolean matrices • Learn the relationship between Boolean matrices and different closures of a relation • Explore how to find the transitive closure using Warshall’s algorithm Discrete Mathematical Structures: Theory and Applications

  3. Matrices Discrete Mathematical Structures: Theory and Applications

  4. Matrices Discrete Mathematical Structures: Theory and Applications

  5. Matrices Discrete Mathematical Structures: Theory and Applications

  6. Matrices Discrete Mathematical Structures: Theory and Applications

  7. Matrices Discrete Mathematical Structures: Theory and Applications

  8. Matrices • Two matrices are added only if they have the same number of rows and the same number of columns • To determine the sum of two matrices, their corresponding elements are added Discrete Mathematical Structures: Theory and Applications

  9. Matrices Discrete Mathematical Structures: Theory and Applications

  10. Matrices Discrete Mathematical Structures: Theory and Applications

  11. Matrices Discrete Mathematical Structures: Theory and Applications

  12. Matrices Discrete Mathematical Structures: Theory and Applications

  13. Matrices • The multiplication AB of matrices A and B is defined only if the number of rows and columns of A is the same as the number of rows and of B Discrete Mathematical Structures: Theory and Applications

  14. Matrices Figure 4.1 • Let A = [aij]m×nbe an m × n matrix and B = [bjk]n×pbe an n × p matrix. Then AB is defined • To determine the (i, k)th element of AB, take the ith row of A and the kth column of B, multiply the corresponding elements, and add the result • Multiply corresponding elements as in Figure 4.1 Discrete Mathematical Structures: Theory and Applications

  15. Discrete Mathematical Structures: Theory and Applications

  16. Matrices Discrete Mathematical Structures: Theory and Applications

  17. Discrete Mathematical Structures: Theory and Applications

  18. Matrices • The rows of A are the columns of ATand the columns of A are the rows of AT Discrete Mathematical Structures: Theory and Applications

  19. Discrete Mathematical Structures: Theory and Applications

  20. Matrices • Boolean (Zero-One) Matrices • Matrices whose entries are 0 or 1 • Allows for representation of matrices in a convenient way in computer memory and for design and implement algorithms to determine the transitive closure of a relation Discrete Mathematical Structures: Theory and Applications

  21. Matrices • Boolean (Zero-One) Matrices • The set {0, 1} is a lattice under the usual “less than or equal to” relation, where for all a, b ∈ {0, 1}, a ∨ b = max{a, b} and a ∧ b = min{a, b} Discrete Mathematical Structures: Theory and Applications

  22. Matrices Discrete Mathematical Structures: Theory and Applications

  23. Matrices Discrete Mathematical Structures: Theory and Applications

  24. Matrices Discrete Mathematical Structures: Theory and Applications

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