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Matrices

Matrices. Section 2.6 Monday, 2 June. Outline. Introduction Matrix Arithmetic: Sum, Product Transposes and Powers of Matrices Identity matrix, Transpose, Symmetric matrices Zero-one Matrices: Join, Meet, Boolean product. 2. Introduction.

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Matrices

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  1. Matrices Section 2.6 Monday, 2 June MATRICES

  2. Outline • Introduction • Matrix Arithmetic: • Sum, Product • Transposes and Powers of Matrices • Identity matrix, Transpose, Symmetric matrices • Zero-one Matrices: • Join, Meet, Boolean product 2 MATRICES

  3. Introduction Definition: If m and n are positive integers, an m×n matrix consists of (mn) elements, arranged in m rows and n columns. element in ith row, jth column m rows n columns Also written as A=aij When m = n, Ais called asquare matrix. 3 MATRICES

  4. Matrix Equality • Definition: Let Aand B be two matrices. Then A= B if A and B have the same number of rows and the same number of columns, and every element at each position in A equals the element at corresponding position in B. * Showing equality is not trivial if elements are real numbers subject to digital approximation. 4 MATRICES

  5. Matrix Addition, Subtraction Let A = aij, B = bij be mn matrices. Then: A + B = aij + bij, and A–B= aij–bij 5 MATRICES

  6. Inventories • Makealot, Inc. manufactures widgets, nerfs, smores, and flots. • It supplies three different warehouses (#1,#2,#3). Opening inventory: Sales: Closing inventory: wi ne sm fl #1 – = #2 #3 MATRICES

  7. MATLAB is “the matrix lab” >> A = [ 1, -1; 3, 4; 2, 0 ] A = 1 -1 3 4 2 0 >> B = [ 3, 4; 1, -4; 2, 3] B = 3 4 1 -4 2 3 >> A+B ans = 4 3 4 0 4 3 >> A-B ans = -2 -5 2 8 0 -3 Octave is a free version from Source Forge MATRICES

  8. Matrix Multiplication Let A be an mk matrix, and B be a kn matrix. Then their product is: AB=[cij], where 8 MATRICES

  9. Matrix Multiplication Let A be an mk matrix, and B be a kn matrix. Then their product is: AB=[cij], where 9 MATRICES

  10. Matrix Multiplication + + = a b a b a b c 21 12 22 22 23 32 22 Let A be an mk matrix, and B be a kn matrix. Then their product is: AB=[cij] 10 MATRICES

  11. Example >> A A = 1 -1 3 4 2 0 >>C = 3 1 2 4 -4 3 >> A*C ans = -1 5 -1 25 -13 18 6 2 4 % to get ans row 2, col 3, % dot product A row 2 by % C col 3 >> [ 3, 4] * [2; 3] ans = 18 >> dot ( [3,4], [2, 3]) ans = 18 Note the semicolon to produce a 2 x1 column vector or matrix MATRICES

  12. Matching Dimensions • Given an n×mmatrix A and an r×smatrix B: • The product AB is defined only if m = r . • When defined, AB is an n×smatrix. 23 34 24 12 MATRICES

  13. Nutrient counting - EXERCISE • Construct:F: food×nutrient matrixC: day× units-consumed matrix • Using F and C, calculate:W: day×nutrient matrix • What do the following represent? • W(1,1)? • W(1,2)? • W(2, 3)? MATRICES

  14. Nutrient counting F: C:  W: = = W(1,1): 355 Fat Cal’s on Sat; W(1,2): 220 Prot Cal’s on Sat;W(1,3): 1250 Sodium on Sat; W(2,1): 330 Fat Cal’s on Sun; etc. MATRICES

  15. Nutrition in combined food units >> Food = [25, 30, 100; 50, 10, 250; 40, 20, 150; 40, 15, 100 ] Food = 25 30 100 50 10 250 40 20 150 40 15 100 >> Consumption = [3 0 5 2; 5, 1, 0, 4] Consumption = 3 0 5 2 5 1 0 4 >> Consumption * Food ans = 355 220 1250 335 220 1150 MATRICES

  16. Multiplicative Properties • Prove or disprove: If A and B are matrices, and AB is defined, then BA is also defined. • This is false. Consider, e.g,: • Prove or disprove: If A and B are matrices and if AB and BA are both defined, then AB = BA. • This is false. Consider, e.g., • Matrix multiplication is not commutative Under what circumstances are ABand BA both defined? 16 MATRICES

  17. Multiplicative Properties • Let A, B and C be matrices. Prove or disprove: If the product (AB)C is defined, then so is the product A(BC). • This is true. To see why, assume that A is m ×n. Then, since AB is defined, there is some positive integer ksuch that B is n ×k and AB is m×k. As (AB)C is defined and AB is m×k, there is some positive integer l such that C is k ×l.These conclusions imply that BC is defined and is n ×l, which in turn implies that A(BC) is defined (and is m ×l). MATRICES

  18. Multiplicative Properties • Let A, B and C be matrices. If (AB)C defined, then (AB)C = A(BC). • Assume A is m ×n, B is n ×k and C is k ×l. The result follows from: ipth element of AB qjth element of BC ijth element of (AB)C ijth element of A(BC) MATRICES

  19. Cost of Matrix Product 23 34 a11b12+ a12b22+ a13b32= c12 It takes 3 multiplications (and 2 additions) to calculate each element. There are 24=8 elements to calculate. So 243 multiplications are needed. In general to multiply an (mk) matrix and a (kn) matrix requires m·k·n multiplications. 19 MATRICES

  20. Best Order? Consider the product of the following matrices. Should we do (AB)C or A(BC)? (A B) C 20 ∙ 40 ∙10 = 8000 operations 32000 20 ∙ 30 ∙40 = 24000 operations A (B C) 30 ∙ 40 ∙10 = 12000 operations 18000 20 ∙ 30 ∙10 = 6000 operations So, A(BC) is bestin this case. 20 MATRICES

  21. Identity Matrix The identity matrix is a square matrix with all 1’s along the diagonal and 0’s elsewhere. We write In to denote the n ×nidentity matrix. Given an mn matrix A, we have A =Im A = A In 21 MATRICES

  22. Identity Matrix The identity matrix is a square matrix with all 1’s along the diagonal and 0’s elsewhere. We write In to denote the n ×nidentity matrix. Given an mn matrix A, we have A =Im A = A In 22 MATRICES

  23. Inverse Matrix • Let A and B be nn matrices. • If AB=BA=Inthen B is called the inverse of A, denoted B=A-1. • Not all square matrices are invertible. Thus: 23 MATRICES

  24. Application: Solving linear equalities • Represent a system of linear equalities as a matrix equation: • There is a unique solution if the coefficient matrix is invertible: MATRICES

  25. Transpose • Given an n ×m matrix A = (aij), the transpose of A, written At, is the m ×n matrix that has element aij in row j and column i, for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Flip across diagonal MATRICES

  26. Symmetric Matrix A square matrix A is said to be symmetric if A = At. Non-symmetric: Symmetric: 26 MATRICES

  27. Exercise Are the following matrices symmetric? • The 100 × 100 matrix M = [mij] satisfying mij= gcd(i,j) • The 100 × 100 matrix S = [sij] satisfying sij= i−j • I2000 MATRICES

  28. Powers of matrices • Let A be ann ×n matrix. Thepowers of A are defined recursively:0th power of A: A0 = Inkth power of A: Ak = A∙Ak −1, for k > 0 . . . 28 MATRICES

  29. Powers of matrices • Let A be ann ×n matrix. Thepowers of A are defined recursively:0th power of A: A0 = Inkth power of A: Ak = A∙Ak −1, for k > 0 k Ak= AA ∙∙∙ A 29 MATRICES

  30. Exercise Calculate MATRICES

  31. Exercise Calculate MATRICES

  32. Zero-one Matrices • All entries are 0 or 1. • Operations are Boolean operations: Let A = [aij] and B = [bij] be m ×n zero-one matrices. Then: • The join of A and B, written A  B, is the m ×n zero-one matrix [aijbij]. • The meet of A and B, written A ∧B, is the m ×n zero-one matrix [aij∧bij]. 32 MATRICES

  33. Flight connections Delta connection matrix: Delta operates direct flights: From: To: From: To DTW JFK JFK DTW JFK MIA MIA JFK MIA LAX LAX MIA Airtran operates direct flights: From: To: From: To: FNT DTW DTW CLE CLE TOL TOL JFK DTW JFK DTW MIA Order the airport codes: 1:CLE, 2:DTW, 3:FNT, 4:JFK, 5:LAX, 6:MIA, 7: TOL C D F J L M T D = C D F J L M T MATRICES

  34. Flight connections Delta operates direct flights: From: To: From: To DTW JFK JFK DTW JFK MIA MIA JFK MIA LAX LAX MIA Airtran operates direct flights: From: To: From: To: FNT DTW DTW CLE CLE TOL TOL JFK DTW JFK DTW MIA Order the airport codes: 1:CLE, 2:DTW, 3:FNT, 4:JFK, 5:LAX, 6:MIA, 7: TOL Airtran connections: C D F J L M T A = C D F J L M T MATRICES

  35. Flight connections - EXERCISE C D F J L M T C D F J L M T D = A = C D F J L M T C D F J L M T What operation on A and D produces the matrix showing the airports connected by a Delta flight or an Airtran flight? What operation on A and D produces the matrix showing the airports connected by both a Delta flight and an Airtranflight? (You have a choice of airlines.) MATRICES

  36. Flight connections Airports connected by a Delta flight or an Airtran flight: D  A =  = MATRICES

  37. Flight connections Airports connected by both a Delta flight and an Airtran flight: D  A =  = MATRICES

  38. MATLAB computations >> or(Delta, Airtran) ans = 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 >> and(Delta, Airtran) ans = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 >> Delta = [ 0 0 0 0 0 0 0; 0 0 0 1 0 0 0; 0 0 0 0 0 0 0; 0 1 0 0 0 1 0; 0 0 0 0 0 1 0; 0 0 0 1 1 0 0; 0 0 0 0 0 0 0 ]; >> Airtran = [ 0 0 0 0 0 0 1; 1 0 0 1 0 1 0; 0 1 0 0 0 0 0; 0 0 0 0 0 0 0; 0 0 0 0 0 0 0; 0 0 0 0 0 0 0; 0 0 0 1 0 0 0 ]; MATRICES

  39. Boolean Product AB Domination law says you can stop when you find a ‘1’ AB 39 MATRICES

  40. Boolean Product Properties AB  BA In general, A⊙B  B⊙A Example: BA AB 40 MATRICES

  41. Boolean Power • A Boolean power matrix can be defined in exactly the same way as a power matrix. For a nn square matrix A, the power matrix: A[0] = In and A[r] = AA[r −1] for r > 0 AA  . . .  A 41 MATRICES

  42. Flight connections Airports connected via taking a Delta flight and then an Airtran flight. DA =  = C D F J L M T C D F J L M T C D F J L M T MATRICES

  43. Flight connections Airports connected via taking a Delta flight and then an Airtran flight. DA =  = C D F J L M T C D F J L M T C D F J L M T MATRICES

  44. Flight connections Airports connected via taking a Delta flight and then an Airtran flight. DA =  = C D F J L M T C D F J L M T C D F J L M T MATRICES

  45. Flight connections - EXERCISE Calculate the airports connected via taking an Airtran flight and then a Delta flight: AD =  = C D F J L M T MATRICES

  46. Flight connections Airports connected via any number of Delta or Airtran flights (including no flights). I (DA)  (DA)[2] (DA)[3] (DA)[4] … From CLE: get to all but FNT. . .From FNT: get to all. . . To JFK: from any of the airports . . . To FNT: only from FNT C D F J L M T MATRICES

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