CSCI 171
Presentation 9
Matrix Theory
Matrix Theory
• Matrix – Rectangular array
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ith row, jth column, i,j element
Square matrix, diagonal
Diagonal matrix
Equality
Zero Matrix (additive identity)
Identity Matrix (multiplicative identity)
Matrix Theory
• Addition
• Theorem 1
– i) A + B = B + A
– ii) (A + B) + C = A + (B + C)
– iii) A + 0 = 0 + A = A
Matrix Theory
• Multiplication
• Theorem 2
– i) A(BC) = (AB)C
– ii) A(B + C) = AB + AC
– iii) (A + B)C = AC + BC
Matrix Theory
• Commutativity of Multiplication?
• Let A be size m x p, B be size p x n
• BA:
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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
• Other properties / definitions:
– If A is m x n, then ImA = AIn = A
– If A is square (n x n):
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Ap = AAA…A (p factors)
A0 = In
ApAq = A(p+q)
(Ap)q = Apq
– (AB)p = ApBp if and only if AB = BA
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
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
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 x n)
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
Matrix Theory
• 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)