A very brief introduction to Matrix Section 2

A very brief introduction to Matrix (Section 2. 7) • Definitions • Some properties • Basic matrix operations • Zero-One (Boolean) matrices

Matrix (Section 2. 7) Definition: A matrix is a rectangular array of numbers. A matrix of m rows and n columns is called an m n matrix, denoted Am n. The element or entry at the ith row and jth column is denoted ai, j. The matrix can also be denoted A = [ai, j]. Example row column a 2, 3 = 2

Matrix Two matrices Am n and Bp q are equal if they have the same number of rows and columns (m = p and n = q), and their corresponding entries are equal (ai, j =bi, j for all i, j). Am n is a square matrix if m = n, denoted Am A square matrix A is said to be symmetric if ai, j = aj, i for all i and j.

Matrix arithmetic (operations) Matrix addition. Am n and Bm n • must have the same numbers of rows and columns • add corresponding entries Am n + Bm n = Cm n = [ai, j + bi, j] Matrix subtraction is done similarly

Matrix arithmetic (operations) Multiply a matrix by a number. • b A = [b ai, j] (i. e. , multiply the number to each entry. ) Multiplication of two matrices. Am k and Bk n • number of columns of the first must equal number of rows of the second • the product is a matrix, denoted AB = Cm n • Entry ci, j is the sum of pair-wise products of the ith row of A and jth column of B

Matrix arithmetic (operations) Example

Powers and Transposes Identity matrix: In • A square matrix of n rows and n columns • Diagonal entries are 1, all other entries are 0 (ii, i = 1 for all i, ii, j = 0 for all i != j. ) • For matrix Am n, we have Im A = A In = A Powers of (square) matrix An A 0 = In =, Ar = AA···A r times

Powers and Transposes Matrix transpose: Am n • the transpose of A, denoted At, is a n m matrix • At = [bi, j = aj, i] • ith row of A becomes ith column of At Theorem: A square matrix An is symmetric iff A = At

Zero-One (Boolean) Matrix Definition: • Entries are Boolean values (0 and 1) • Operations are also Boolean Matrix join. • A B = [ai, j bi, j] Example: Matrix meet. • A B = [ai, j bi, j]

Zero-One (Boolean) Matrix multiplication: Am k and Bk n • the product is a Zero-One matrix, denoted A B = Cm n • cij = (ai 1 b 1 j) (ai 2 b 2 i) … (aik bkj). Example: