1. CSCI 171 Presentation 9 Matrix Theory

2. 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)

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

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

5. 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 Matrix Theory

6. Other properties / definitions: –If A is m x n, then I m A = AI n = A –If A is square (n x n): A p = AAA … A (p factors) A 0 = I n A p A q = A (p+q) (A p ) q = A pq –(AB) p = A p B p if and only if AB = BA Matrix Theory

7. Transposition Theorem 3 –i) (A t ) t = A –ii) (A + B) t = A t + B t –iii) (AB) t = B t A t Symmetry (A t = A) –A is symmetric if and only if a i,j = a j,i for all i and j Matrix Theory

8. 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): C i,j = 1 if A i,j = 1 or B i,j = 1 C i,j = 0 if A i,j = 0 and B i,j = 0 –The meet of A and B (C = A  B): C i,j = 1 if A i,j = 1 and B i,j = 1 C i,j = 0 if A i,j = 0 or B i,j = 0 Matrix Theory

9. 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 x n ) is (C = A  B): C i,j = 1 if A i,j =1 and B k,j = 1 for some k, 1  k  p C i,j = 0 otherwise Matrix Theory

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