Mathematics 13: Lecture 10

Transcription

1 Mathematics 13: Lecture 10 Matrices Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

2 Matrices Recall: A matrix is a rectangular array of numbers. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

3 Matrices Recall: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is an m n matrix. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

4 Matrices Recall: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is an m n matrix. The number in the ith row and jth column of a matrix is the (i, j)-entry of the matrix. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

5 Examples A 2 3 matrix: [ ] Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

6 Examples A 2 3 matrix: [ ] A 4 2 matrix: Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

7 Notation We may denote a general m n matrix A as a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n A = a m1 a m2 a m3 a mn Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

8 Notation We may denote a general m n matrix A as a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n A = a m1 a m2 a m3 a mn For a more compact notation, we will often write simply A = [a ij ]. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

9 Algebraic operations If A = [a ij ] and B = [b ij ] are both m n matrices and c is a scalar, then we define Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

10 Algebraic operations If A = [a ij ] and B = [b ij ] are both m n matrices and c is a scalar, then we define A + B = [aij + b ij ], Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

11 Algebraic operations If A = [a ij ] and B = [b ij ] are both m n matrices and c is a scalar, then we define A + B = [aij + b ij ], ca = [ca ij ]. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

12 Algebraic operations If A = [a ij ] and B = [b ij ] are both m n matrices and c is a scalar, then we define A + B = [aij + b ij ], ca = [ca ij ]. That is, we add two matrices of the same size by adding their respective entries, term by term, and we multiply a matrix by a scalar by multiplying each entry by the scalar. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

13 Algebraic operations If A = [a ij ] and B = [b ij ] are both m n matrices and c is a scalar, then we define A + B = [aij + b ij ], ca = [ca ij ]. That is, we add two matrices of the same size by adding their respective entries, term by term, and we multiply a matrix by a scalar by multiplying each entry by the scalar. Note: This generalizes addition and scalar multiplication of vectors. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

14 Examples Let A = 3 4 and B = Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

15 Examples Let Then A = 3 4 and B = A + B = and 3A = Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

16 Transposition If A = [a ij ] is an m n matrix, then we call the n m matrix the transpose of A. A T = [b ij ] where b ij = a ji Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

17 Transposition If A = [a ij ] is an m n matrix, then we call the n m matrix the transpose of A. A T = [b ij ] where b ij = a ji That is, we obtain A T by reflecting A across its diagonal entries (the entries a 11, a 22,..., a mm ). Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

18 Transposition If A = [a ij ] is an m n matrix, then we call the n m matrix the transpose of A. A T = [b ij ] where b ij = a ji That is, we obtain A T by reflecting A across its diagonal entries (the entries a 11, a 22,..., a mm ). That is, the columns of A are the rows of A T and the rows of A are the columns of A T. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

19 Examples If 2 3 A = 1 0, then A T = 5 4 [ 2 1 ] Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

20 Examples If If 2 3 A = 1 0, then A T = 5 4 [ 2 1 ] B = 2, then B T = [ ]. 3 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

21 Examples If If 2 3 A = 1 0, then A T = 5 4 [ 2 1 ] B = 2, then B T = [ ]. 3 Note: If B is a column matrix, then B T is a row matrix (and vice versa). Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

22 Symmetric matrices We say a matrix A is symmetric if A T = A. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

23 Symmetric matrices We say a matrix A is symmetric if A T = A. A symmetric matrix is necessarily a square matrix, that is, a matrix with the same number of rows and columns. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

24 Symmetric matrices We say a matrix A is symmetric if A T = A. A symmetric matrix is necessarily a square matrix, that is, a matrix with the same number of rows and columns. Examples: If A = [ ] and B = then A is symmetric and B is not symmetric. [ ] 1 2, 3 4 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

25 Some Octave Let A = [ ] and B = [ ] Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

26 Some Octave Let A = [ ] and B = [ ] Octave commands to input A and B: A = [1 2 3; ] B = [-1 3 8; ] Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

27 Some Octave Let A = [ ] and B = [ ] Octave commands to input A and B: A = [1 2 3; ] B = [-1 3 8; ] Octave commands to compute A + B, 4B, and 4A 7B: A + B 4*B 4*A - 7*B Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

28 Some Octave Let A = [ ] and B = [ ] Octave commands to input A and B: A = [1 2 3; ] B = [-1 3 8; ] Octave commands to compute A + B, 4B, and 4A 7B: A + B 4*B 4*A - 7*B To find A T : A Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

29 Example Two companies, A and B, each purchase three different types of widgets, α, β, and γ, to manufacture their respective products. They may purchase their widgets from one of two suppliers, I and II. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

30 Example Two companies, A and B, each purchase three different types of widgets, α, β, and γ, to manufacture their respective products. They may purchase their widgets from one of two suppliers, I and II. The following two arrays specify the demand that each company has for each type of widget and the price each supplier charges for the respective types of widgets: Widget α Widget β Widget γ Company A Compnay B Supplier I Supplier II Widget α Widget β Widget γ Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

31 Example (cont d) The cost for Company A to purchase widgets from Supplier I is then = and the cost from Supplier II is = Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

32 Example (cont d) The cost for Company A to purchase widgets from Supplier I is then = and the cost from Supplier II is = For Company B, the cost from Supplier I is = and from Supplier II is = Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

33 Example (cont d) The cost for Company A to purchase widgets from Supplier I is then = and the cost from Supplier II is = For Company B, the cost from Supplier I is = and from Supplier II is = Note: These are the dot products of the rows of the first array with the columns of the second array. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

34 Example (cont d) Note: we may also present the preceding costs in an array: Suppier I Supplier II Company A Company B Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

35 Example (cont d) Note: we may also present the preceding costs in an array: Suppier I Supplier II Company A Company B Now let D be the demand matrix formed from our first array, P be the price matrix formed from our second array, and C be the cost matrix formed from the last array. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

36 Example (cont d) That is, let D = and C = [ ] , P = [ ] , Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

37 Example (cont d) That is, let D = and C = [ ] , P = [ ] , If we let D = [d ij ], P = [p ij ], and C = [c ij ], then we have c 11 = d 11 p 11 + d 12 p 21 + d 13 p 31, c 21 = d 21 p 11 + d 22 p 21 + d 23 p 31, c 12 = d 11 p 12 + d 12 p 22 + d 13 p 32, c 22 = d 21 p 12 + d 22 p 22 + d 23 p 32. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

38 Example (cont d) That is, let D = and C = [ ] , P = [ ] , If we let D = [d ij ], P = [p ij ], and C = [c ij ], then we have c 11 = d 11 p 11 + d 12 p 21 + d 13 p 31, c 21 = d 21 p 11 + d 22 p 21 + d 23 p 31, c 12 = d 11 p 12 + d 12 p 22 + d 13 p 32, c 22 = d 21 p 12 + d 22 p 22 + d 23 p 32. We will call C the product of D and P and write C = DP. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

39 Matrix product If A = [a ij ] is an m n matrix and B = [b ij ] is an n p matrix, then the product of A and B is the m p matrix whose (i, j)-entry is the dot product of the ith row of A with the jth column of B. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

40 Matrix product If A = [a ij ] is an m n matrix and B = [b ij ] is an n p matrix, then the product of A and B is the m p matrix whose (i, j)-entry is the dot product of the ith row of A with the jth column of B. That is, if C = [c ij ] = AB, then c ij = n a ik b kj = a i1 b 1j + a i2 b 2j + + a in b nj. k=1 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

41 Examples We have [ ] 6 [ ] = Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

42 Examples We have [ ] 6 [ ] = And [ ] [ ] = but [ ] [ ] = [ ] 12 7, 18 2 [ ] Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

43 Examples We have [ ] 6 [ ] = And [ ] [ ] = but [ ] [ ] = [ ] 12 7, 18 2 [ ] Note: AB may not be equal to BA even if both exist and are the same size. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

44 Definition We call the n n matrix I = the identity matrix. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

45 Definition We call the n n matrix I = the identity matrix. If A is m n, then AI n = A. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

46 Definition We call the n n matrix I = the identity matrix. If A is m n, then AI n = A. If B is n p, then I n B = B. Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

47 Octave commands To find the product of matrices A and B: A*B Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

48 Octave commands To find the product of matrices A and B: A*B To find the 10 th power of a square matrix (that is, a matrix with the same number of rows and columns): A^10 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

49 Example The system of equations 3x + 4y 7z = 6 x y + z = 5 may be written as [ ] x y = z [ ] 6. 5 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19

50 Example The system of equations 3x + 4y 7z = 6 x y + z = 5 may be written as [ ] x y = z [ ] 6. 5 Or as A x = b, where A = [ ] 3 4 7, x = x y, and b = z [ ] 6. 5 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, / 19