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Matrices MSU CSE 260 Outline • Introduction • Matrix Arithmetic: – Sum, Product • Transposes and Powers of Matrices – Identity matrix, Transpose, Symmetric matrices • Zero-one Matrices: – Join, Meet, Boolean product • Exercise 2.6 Introduction Definition A matrix is a rectangular array of numbers. a11 a12 a a22 21 A a m1 am 2 a1n a2 n aij amn element in ith row, jth column m rows Also written as A=aij mn matrix n columns When m=n, A is called a square matrix. Matrix Equality • Definition Let A and B be two matrices. A=B if they have the same number of rows and columns, and every element at each position in A equals element at corresponding position in B. Matrix Addition, Subtraction Let A = aij , B = bij be mn matrices. Then: A + B = aij + bij, and A - B = aij - bij 1 1 3 4 4 3 3 4 1 4 4 0 2 0 2 3 4 3 1 1 3 4 2 5 3 4 1 4 2 8 2 0 2 3 0 3 Matrix Multiplication Let A be a mk matrix, and B be a kn matrix, AB cij k cij ait btj ai1b1 j ai 2b2 j ... aik bkj t 1 a11 a 21 a12 a22 a13 a23 b11 b12 b b 21 22 b31 b32 b13 b23 b33 b14 c11 c12 b24 c21 c22 b34 a11b12 a12b22 a13b32 c12 c13 c23 c14 c24 Matching Dimensions To multiply two matrices, inner numbers must match: Otherwise, 23 34 not defined. 24 matrix have to be equal a11 a 21 a12 a22 23 a13 a23 b11 b12 b b 21 22 b31 b32 b13 b23 b33 34 b14 c11 c12 c13 b24 c21 c22 c23 b34 24 c14 c24 Multiplicative Properties Note that just because AB is defined, BA may not be. Example If A is 34, B is 46, then AB=36, but BA is not defined (46 . 34). Even if both AB and BA are defined, they may not have the same size. Even if they do, matrices do not commute. 1 1 2 1 A B 2 1 1 1 3 2 4 3 AB BA 5 3 3 2 however ( AB)C A( BC ), assuming dimensions match Efficiency of Multiplication a11 a 21 23 a12 a13 a22 a23 34 b12 b13 b11 b b22 21 b31 b32 b23 b33 b14 c11 c12 b24 c21 c22 b34 c13 c23 c14 c24 a11b12 + a12b22 + a13b32 = c12 Takes 3 multiplications, and 2 additions for each element. This has to be done 24 (=8) times (since product matrix is 24). So 243 multiplications are needed. •(mk) (kn) matrix product requires m.k.n multiplications. Best Order? Let A be a 2030 matrix, B 3040, C 4010. (AB)C or A(BC)? (2030 3040) 4010 20 40 10 8000 operations 20 30 40 24000 operations 32000 2030 (3040 4010) 30 40 10 12000 operations 20 30 10 6000 operations So, A(BC) is best in this case. 18000 Identity Matrix The identity matrix has 1’s down the diagonal, e.g.: 1 0 0 I 3 0 1 0 0 0 1 1 0 0 a 0 1 0 c 0 0 1 e For a mn matrix A, Im A = A In mm mn = mn nn b a 0 0 b 0 0 d 0 c 0 0 d 0 f 0 0 e 0 0 f Inverse Matrix Let A and B be nn matrices. If AB=BA=In then B is called the inverse of A, denoted B=A-1. Not all square matrices are invertible. Use of Inverse to Solve Equations a1 a2 b b 2 1 c1 c2 a3 x p b3 y q c3 z r a1 x a2 y a3 z p b x b y b z q 2 3 1 c1 x c2 y c3 z r 1 1 x a 1 a 2 y b 11 b 12 1 1 z c 1 c 2 a 13 p b 13 q 1 c 3 r AX K A1 AX A1 K IX A1 K Please note that a-1j is NOT necessarily (aj)-1. Transposes of Matrices a ji aij Flip across diagonal 1 4 1 2 3 2 5 written At 4 5 6 3 6 Transposes are used frequently in various algorithms. Symmetric Matrix If At A 1 4 1 4 3 0 1 0 2 In A is called symmetric. is symmetric. Note, for A to be symmetric, is has to be square. is trivially symmetric... Examples ( AB) t B t At k AB cij aix bxj x 1 k t ( AB) t cij c ji a jx bxi x k k k B t At b t ix a t xj bxi a jx a jx bxi x 1 x 1 x 1 Power Matrix • For a nn square matrix A, the power matrix is defined as: Ar = A A … A r times • A0 is defined as In. Zero-one Matrices • All entries are 0 or 1. • Operations are and . • Boolean product is defined using: for multiplication, and for addition. Boolean Operations 1 0 1 0 1 0 A B 0 1 0 1 1 0 0 A B called "meet" A B 0 1 A B called " join" A B 1 0 1 1 1 0 0 1 0 Terminology is from Boolean Algebra.Think “join” is “put together”, like union, and “meet” is “where they meet”, or intersect. Boolean Product A aij be m k , and B bij be k n A B cij , cij (ai1 b1 j ) (ai 2 b2 j ) (aik bkj ) (Should be a ‘dot’) 1 0 1 1 0 A 0 1 , B 0 1 1 1 0 Since this is “or’d”, you can stop when you find a ‘1’ (1 1) (0 0) 1 0 1 1 0 A B (0 1) (1 0) 1 1 0 1 1 (1 1) (0 0) 1 0 1 1 0 Boolean Product Properties • In general, A B B A • Example 1 1 0 0 0 1 A 0 1 1 , B 0 0 1 1 1 0 1 1 1 0 0 1 A B 1 1 1 0 0 1 1 1 0 B A 1 1 0 1 1 1 Boolean Power • A Boolean power matrix can be defined in exactly the same way as a power matrix. For a nn square matrix A, the power matrix is defined as: A[r] = A A … A r times A[0] is defined as In. Exercise 2.6