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CSCI 171

CSCI 171. Presentation 9 Matrix Theory. Matrix Theory. Matrix – Rectangular array i th row, j th column, i,j element Square matrix, diagonal Diagonal matrix Equality Zero Matrix (additive identity) Identity Matrix (multiplicative identity). Matrix Theory. Addition Theorem 1

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CSCI 171

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  1. CSCI 171 Presentation 9 Matrix Theory

  2. Matrix Theory • Matrix – Rectangular array • ith row, jth column, i,j element • Square matrix, diagonal • Diagonal matrix • Equality • Zero Matrix (additive identity) • Identity Matrix (multiplicative identity)

  3. Matrix Theory • Addition • Theorem 1 • i) A + B = B + A • ii) (A + B) + C = A + (B + C) • iii) A + 0 = 0 + A = A

  4. Matrix Theory • Multiplication • Theorem 2 • i) A(BC) = (AB)C • ii) A(B + C) = AB + AC • iii) (A + B)C = AC + BC

  5. Matrix Theory • Commutativity of Multiplication? • Let A be size m x p, B be size p x n • BA: • May not be defined • May be defined, but a different size than AB • May be defined, same size as AB, but ABBA • May be equal to AB

  6. Matrix Theory • Other properties / definitions: • If A is m x n, then ImA = AIn = A • If A is square (n x n): • Ap = AAA…A (p factors) • A0 = In • ApAq = A(p+q) • (Ap)q = Apq • (AB)p = ApBp if and only if AB = BA

  7. Matrix Theory • Transposition • Theorem 3 • i) (At)t = A • ii) (A + B)t = At + Bt • iii) (AB)t = BtAt • Symmetry (At = A) • A is symmetric if and only if ai,j = aj,i for all i and j

  8. Matrix Theory • Boolean Matrices (all elements are 0 or 1) • Operations on Boolean Matrices: • Let A and B be boolean Matrices • The join of A and B (C = A  B): • Ci,j = 1 if Ai,j = 1 or Bi,j = 1 • Ci,j = 0 if Ai,j = 0 and Bi,j = 0 • The meet of A and B (C = A  B): • Ci,j = 1 if Ai,j = 1 and Bi,j = 1 • Ci,j = 0 if Ai,j = 0 or Bi,j = 0

  9. Matrix Theory • Boolean Matrices (all elements are 0 or 1) • Operations on Boolean Matrices: • Let A and B be boolean Matrices • The boolean product of A (m x p) and B (p xn) is (C = A  B): • Ci,j = 1 if Ai,j =1 and Bk,j = 1 for some k, 1  k  p • Ci,j = 0 otherwise

  10. Matrix Theory • Boolean Matrices (all elements are 0 or 1) • Theorem 4If A, B, and C are boolean matrices of appropriate sizes, then:i) A  B = B  Aii) A  B = B  Aiii) (A  B)  C = A  (B  C)iiii) (A  B)  C = A  (B  C)

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