Introduction Matrices A matrix say MAYtrix is a

Introduction Matrices A matrix (say MAY-trix) is a rectangular array of objects (usually numbers). (행렬은 수의 사각형 배열이다. ) An m n (“m by n”) matrix has exactly m horizontal rows, and n vertical columns. (m개의 행과 n개의 열을 갖는 행렬) Plural of matrix = matrices An n n matrix is called a square matrix, whose order is n. (행과 열의 개수가 같은 행렬을 정방행렬이라 한다. ) Tons of applications: • Models within Computational Science & Engineering • Computer Graphics, Image Processing, Network Modeling • Many, many more … Page 2 Discrete Mathematics by Yang-Sae Moon

Matrix Equality Matrices Two matrices A and B are equal iff they have the same number of rows, the same number of columns, and all corresponding elements are equal. (두 행렬이 같은 수의 행과 열을 가지며 각 위치의 해당 원소의 값이 같으 면 “두 행렬은 같다”고 정의한다. ) Example Page 3 Discrete Mathematics by Yang-Sae Moon

Row and Column Order (1/2) Matrices The rows in a matrix are usually indexed 1 to m from top to bottom. (행은 위에서 아래로 1~m의 색인 값을 갖는다. ) The columns are usually indexed 1 to n from left to right. (열은 왼쪽에서 오른쪽으로 1~n의 색인 값을 갖는다. ) Elements are indexed by row, then column. (각 원소는 행 색인, 열 색인의 순으로 표현한다. ) Page 4 Discrete Mathematics by Yang-Sae Moon

Row and Column Order (2/2) Matrices Let A be m n matrix [ai, j], ith row = 1 1 n matrix [ai, 1 ai, 2 … ai, n], jth column = m 1 matrix Page 5 Discrete Mathematics by Yang-Sae Moon

Matrix Sums Matrices The sum A+B of two m n matrices A, B is the m n matrix given by adding corresponding elements. (A+B는 (i, j)번째 원소로서 ai, j+bi, j를 갖는 행렬이다. ) A+B = C = [ci, j] = [ai, j+bi, j] where A = [ai, j] and B = [bi, j] Example Page 6 Discrete Mathematics by Yang-Sae Moon

Matrix Products (1/2) Matrices For an m k matrix A and a k n matrix B, the product AB is the m n matrix: I. e. , element (i, j) of AB is given by the vector dot product of the ith row of A and the jth column of B (considered as vectors). (AB의 원소 (i, j)는 A의 i번째 열과 B의 j번째 행의 곱이다. ) Page 7 Discrete Mathematics by Yang-Sae Moon

Matrix Products (2/2) Matrices Example Matrix multiplication is not commutative! • A = m n matrix, B = r s matrix • AB can be defined when n = r • BA can be defined when s = m • Both AB and BA can be defined when m = n = r = s Page 8 (교환법칙 성립 안 함) Discrete Mathematics by Yang-Sae Moon

Matrix Multiplication Algorithm Matrices procedure matmul(matrices A: m k, B: k n) for i : = 1 to m for j : = 1 to n begin ci, j : = 0 (m)· (n)· ( What’s the of its time complexity? (1)+ for q : = 1 to k ci, j : = ci, j + ai, qbq, j (k) · Answer: (mnk) (1)) end {C=[ci, j] is the product of A and B} Page 9 Discrete Mathematics by Yang-Sae Moon

Identity Matrices (항등 행렬) Matrices The identity matrix of order n, In, is the order-n matrix with 1’s along the upper-left to lower-right diagonal and 0’s everywhere else. ((i, i)번째 원소가 1이고, 나머지는 모두 0인 행렬) AIn = In. A = A Page 10 Discrete Mathematics by Yang-Sae Moon

Matrix Inverses (역행렬) Matrices For some (but not all) square matrices A, there exists a unique multiplicative inverse A-1 of A, a matrix such that A -1 A = I. (정방 행렬 A에 대해서 하나의 유일한 역행렬 A-1이 존재한다. ) n If the inverse exists, it is unique, and A-1 A = AA-1. A -1 A Page 11 I 3 Discrete Mathematics by Yang-Sae Moon

Matrix Transposition (전치 행렬) Matrices If A=[ai, j] is an m n matrix, the transpose of A (often written At or AT) is the n m matrix given by At = B = [bi, j] = [aj, i] (1 i n, 1 j m) Flip across diagonal Page 12 Discrete Mathematics by Yang-Sae Moon

Symmetric Matrices (대칭 행렬) Matrices A square matrix A is symmetric iff A=At. I. e. , i, j n: aij = aji. Which is symmetric? Page 13 Discrete Mathematics by Yang-Sae Moon

Powers of Matrices (멱행렬) Matrices If A is an n n square matrix and p 0, then: Ap AAA···A (A 0 In) p times Example: Page 14 Discrete Mathematics by Yang-Sae Moon

Zero-One Matrices (0 -1 행렬) Matrices Useful for representing other structures. • E. g. , relations, directed graphs (later in course) All elements of a zero-one matrix are 0 or 1 • Representing False & True respectively. The meet of A, B (both m n zero-one matrices): A B : [aij bij] (= [aij bij]) The join of A, B: A B : [aij bij] Page 15 Discrete Mathematics by Yang-Sae Moon

Boolean Products (부울 곱) (1/2) Matrices Let A=[aij] be an m k zero-one matrix, & let B=[bij] be a k n zero-one matrix, The Boolean product of A and B is like normal matrix multiplication, but • using “ ” instead “+” • using “ ” instead of “∙” A⊙B Page 16 Discrete Mathematics by Yang-Sae Moon

Boolean Products (부울 곱) (2/2) Matrices Example: ⊙ Algorithm of Boolean Product procedure Boolean product(A, B: zero-one matrices) for i : = 1 to m for j : = 1 to n begin cij : = 0 for q : = 1 to k cij : = cij (aiq bqj) end {C = [cij] is the Boolean product of A and B. } Page 17 Discrete Mathematics by Yang-Sae Moon

Boolean Powers (부울 거듭제곱) Matrices For a square zero-one matrix A, and any k 0, the kth Boolean power of A is simply the Boolean product of k copies of A. (A의 k 부울린 거듭제곱) A[k] A⊙A⊙…⊙A, A[0] In k times Example: ⊙ ⊙ ⊙ Page 18 Discrete Mathematics by Yang-Sae Moon

Homework #4 Matrices Page 19 Discrete Mathematics by Yang-Sae Moon