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Matrices Notes

Matrices Notes - PSfrag replacements u v i j u v A B 2v-v...

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17 2 Matrices 2.1 Why matrices? Sometimes it would be useful to consider rectangular arrays of numbers as a single entity. Con- sider for example the following equations. 2 x - 3 y = 1 , 2 x - 3 y = - 4 - 4 x + 5 y = 3 , - 4 x + 5 y = 1 The coefficients are the same and the only part that changes is the right-hand side. In a sense we are dealing with only one equation identified by the array of the four coefficients: 2 - 3 - 4 5 Another example is given by the rotation of a Cartesian coordinatesystem. If we rotate a refer- ence frame by an angle θ , the new coordinates ( x 1 , y 1 ) are related to the old ones ( x, y ) by: P x x y y x 0 x 0 y 0 y 0 φ For any pair of points ( x, y ) we can apply this formula to obtain the new coordinates ( x 1 , y 1 ) . The essential four coefficients can be written in a rectangular block. cos θ sin θ - sin θ cos θ

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18 2 MATRICES 2.2 What are matrices? An m × n matrix A is a rectangular array of mn numbers arranged in m rows and n columns (note the order!). If a ij represents the number (element) in the i th row and j th column, then A = a 11 a 12 a 13 . . . a 1 n a 21 a 22 a 23 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 a m 3 . . . a mn Shorthand notation: A = ( a ij ) 1 6 i 6 m rows , 1 6 j 6 n columns Examples: A = 1 2 3 4 5 6 is a 2 × 3 matrix B = 7 8 9 10 11 12 is a 3 × 2 matrix C = 1 - 1 4 - 15 is a 2 × 2 “square” matrix D = a b c d is a 4 × 1 “column” matrix E = ( 5 7 9 12 ) is a 1 × 4 “row” matrix An extremely important matrix is the square identity matrix I n . I 2 = 1 0 0 1 , I 3 = 1 0 0 0 1 0 0 0 1 , I n = 1 0 . . . 0 0 1 . . . 0 . . . . . . . . . 0 0 . . . 1 A matrix can be considered as a group of rows or column vectors. Example : F = 1 2 - 1 4 - 2 - 3
2.3 Operations on matrices? 19 May be written as 2 row vectors or 3 column vectors ( 1 2 - 1 ) , ( 4 - 2 - 3 ) , or 1 4 , 2 - 2 , - 1 - 3 2.3 Operations on matrices? What sort of operations can be performed on matrices? 2.3.1 Transposition The transpos e of an m × n matrix A is an n × m matrix, denoted by A T , whose rows are the columns of A . Example : A = 1 2 3 4 5 6 , A T = 1 4 2 5 3 6 B = 1 2 2 1 , B T = 1 2 2 1 Note: B = B T , i.e. b ij = b ji since B is a symmetric matrix. 2.3.2 Sum of two matrices Only matrices which have the same number of rows and columns can be summed. For examples it’s possible to sum a 5 × 3 matrix with another 5 × 3 matrix, but not with a 4 × 3 one. The sum of two m × n matrices A = ( a ij ) and B = ( b ij ) is a m × n matrix C whose elements c ij are given by: c ij = a ij + b ij

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20 2 MATRICES Example 1: A = 1 2 3 4 5 6 , B = - 1 3 - 1 0 4 3 both 2 × 3 matrices so we can add them C = A + B = 1 - 1 2 + 3 3 - 1 4 + 0 5 + 4 6 + 3 = 0 5 2 4 9 9 2.3.3 Product of a matrix and a number Let t be a real number and A an m × n matrix. Their product is an m × n matrix B obtained by multiplying each element of A by t . b ij = ta ij Example : A = 1 2 3 4 5 6 , t = 4 So that B = 4 A = 4 . 1 4 . 2 4 . 3 4 . 4 4 . 5 4 . 6 = 4 8 12 16 20 24 2.3.4 Product of two matrices If A = ( a ij ) is an m × n matrix and B = ( b ij ) is an n × p matrix, then their product AB is the m × p matrix C = ( c ij ) with elements given by c ij = n X k =1 a ik b kj That is c ij
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