Matrices Notes - 17 2 Matrices 2.1 Why matrices? Sometimes...

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Unformatted text preview: 17 2 Matrices 2.1 Why matrices? Sometimes it would be useful to consider rectangular arrays of numbers as a single entity. Con- sider forexamplethefollowingequations. 2 x- 3 y = 1 , 2 x- 3 y =- 4- 4 x + 5 y = 3 ,- 4 x + 5 y = 1 Thecoefficientsarethesameandtheonlypartthatchangesistheright-handside. Inasensewe are dealingwithonlyone equationidentifiedbythe array of thefourcoefficients: 2- 3- 4 5 Another example is given by the rotation of a Cartesian coordinatesystem. If we rotate a refer- enceframe byan angle , thenew coordinates ( x 1 , y 1 ) are relatedtothe oldones ( x, y ) by: P x x y y x x y y For any pair of points ( x, y ) we can apply this formula to obtain the new coordinates ( x 1 , y 1 ) . The essentialfourcoefficientscan be writteninarectangularblock. cos sin - sin cos 18 2 MATRICES 2.2 Whatare matrices? An m n matrix A is a rectangular array of mn numbers arranged in m rows and n columns (note theorder!). If a ij representsthe number(element) inthe 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 Shorthandnotation: 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 isa 3 2 matrix C = 1- 1 4- 15 isa 2 2 squarematrix D = a b c d isa 4 1 columnmatrix E = ( 5 7 9 12 ) isa 1 4 row matrix An extremelyimportantmatrixisthe squareidentitymatrix I n . I 2 = 1 1 , I 3 = 1 1 1 , I n = 1 . . . 1 . . . . . . . . . . . . . . . 1 A matrixcan be consideredas agroupof rowsorcolumnvectors. Example: F = 1 2- 1 4- 2- 3 2.3 Operationsonmatrices? 19 Maybe written as2 rowvectorsor3 columnvectors ( 1 2- 1 ) , ( 4- 2- 3 ) , or 1 4 , 2- 2 ,- 1- 3 2.3 Operations on matrices? What sortof operationscan be performedonmatrices? 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 columnsof 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 isa symmetric matrix. 2.3.2 Sumoftwomatrices Onlymatriceswhichhavethesamenumberofrowsandcolumnscanbesummed. Forexamples itspossibleto suma 5 3 matrixwithanother 5 3 matrix,butnotwitha 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 givenby: c ij = a ij + b ij 20 2 MATRICES Example1: A = 1 2 3 4 5 6 , B =- 1 3- 1 4 3 both 2 3 matrices sowe can addthem C = A + B = 1- 1 2 + 3 3- 1 4 + 0 5 + 4 6 + 3 = 5 2 4 9 9 2.3.3 Productofamatrixandanumber Let t bea realnumberand A an m n matrix. Their productisan m n matrix B obtainedby multiplyingeachelement of A by t ....
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Matrices Notes - 17 2 Matrices 2.1 Why matrices? Sometimes...

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