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Typology: Summaries
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Dept. Information & Computer Sci., University of Hawaii Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by McGraw-Hill
3.8 Matrices
Applications of Matrices n Tons of applications, including: n Solving systems of linear equations n Computer Graphics, Image Processing n Games n Models within many areas of Computational Science & Engineering n Quantum Mechanics, Quantum Computing n Many, many more…
Row and Column Order n The rows in a matrix are usually indexed 1 to m from top to bottom. n The columns are usually indexed 1 to n from left to right. n Elements are indexed by row, then by column.
m m mn n n ij
1 2 21 22 2 11 12 1
Matrix Sums n The sum A + B of two matrices A , B (which must have the same number of rows, and the same number of columns) is the matrix (also with the same shape) given by adding corresponding elements of A and B. A + B = [ a ij
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − 11 5 11 9 11 3 9 3 0 8 2 6
n For an m × k matrix A and a k × n matrix B , the product AB is the m × n matrix: n I.e. , the element of AB indexed ( i , j ) is given by the vector dot product of the i- th row of A and the j- th column of B (considered as vectors). n Note: Matrix multiplication is not commutative!
=
k ij i j i j ik kj i j ij
1 1 1 2 2
Matrix Product Example ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 1 1 2 1 3 1 1 1 0 2 A and B n Because A is a 2× 3 matrix and B is a 2 × 2 matrix, the product AB is not defined.
University of HawaiiUniversity of Hawaii Matrix Multiplication: Non-Commutative n Matrix multiplication is not commutative! n A : m × n matrix and B : r × s matrix n AB is defined when n = r n BA is defined when s = m n When both AB and BA are defined, generally they are not the same size unless m = n = r = s n If both AB and BA are defined and are the same size, then A and B must be square and of the same size n Even when A and B are both n × n matrices, AB and BA are not necessarily equal
Matrix Multiplication Algorithm procedure matmul (matrices A : m × k , B : k × n ) for i := 1 to m for j := 1 to n begin c ij
for q := 1 to k c ij := _c ij
n n Identity Matrices n The identity matrix of order n, I n , is the rank- n square matrix with 1’s along the upper-left to lower-right diagonal, and 0’s everywhere else. ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎩ ⎨ ⎧ ≠ = = = 0 0 1 0 1 0 1 0 0 0 if 1 if [ ] i j i j n ij I δ Kronecker Delta ∀ 1 ≤ i , j ≤ n
Powers of Matrices If A is an n × n square matrix and p ≥ 0, then: n A p = AAA···A (and A 0 = I n
n Example: p times ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − ⋅ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⋅ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⋅ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − 3 2 4 3 2 1 3 2 1 0 2 1 1 0 2 1 1 0 2 1 1 0 2 1 1 0 2 1 3
Matrix Transposition n If A = [ a ij ] is an m × n matrix, the transpose of A (often written A t or A T ) is the n × m matrix given by A t = B = [ b ij ] = [ a ji ] (1 ≤ i ≤ n ,1 ≤ j ≤ m ) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − 3 2 1 1 2 0 0 1 2 2 1 3 t
Zero-One Matrices n Useful for representing other structures. n E.g. , relations, directed graphs (later on) n All elements of a zero-one matrix are either 0 or 1. n E.g., representing False & True respectively. n The join of A , B (both m × n zero-one matrices): n A ∨ B = [ a ij ∨ b ij
n The meet of A , B : n A ∧ B = [ a ij ∧ b ij ] = [ a ij b ij
Join and Meet Example ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∨ ∨ ∨ ∨ ∨ ∨ ∨ = 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 1 0 A B ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∧ ∧ ∧ ∧ ∧ ∧ ∧ = 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 A B and ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 1 1 0 0 1 0 0 1 0 1 0 1 A B
