1Matrix AlgebraHGEN619 class 2006Heuristicn You already know a lot of itn Economical and aestheticn Great for statistics2What you known All about (1x1) matricesn Operation Example Resultn Addition 2 + 2n Subtraction 5 - 1n Multiplication 2 x 2n Division 12 / 3What you known All about (1x1) matricesn Operation Example Resultn Addition 2 + 2 4n Subtraction 5 – 1 4n Multiplication 2 x 2 4n Division 12 / 3 43What you may guessn Numbers can be organized in boxes, e.g.1423What you may guessn Numbers can be organized in boxes, e.g.14234Matrix NotationAMany Numbers31 23 16 99 08 12 14 73 85 98 33 94 12 75 02 57 92 75 1128 39 57 17 38 18 38 65 10 73 16 73 77 63 18 56 18 57 0274 82 20 10 75 84 19 47 14 11 84 08 47 57 58 49 48 28 4288 84 47 48 43 05 61 75 98 47 32 98 15 49 01 38 65 81 6843 17 65 21 79 43 17 59 41 37 59 43 17 97 65 41 35 54 4475 49 03 86 93 41 76 73 19 57 75 49 27 59 34 27 59 34 8243 19 74 32 17 43 92 65 94 13 75 93 41 65 99 13 47 56 3475 83 47 48 73 98 47 39 28 17 49 03 63 91 40 35 42 12 5431 87 49 75 48 91 37 59 13 48 75 94 13 75 45 43 54 32 5375 48 90 37 59 37 59 43 75 90 33 57 75 89 43 67 74 73 1034 92 76 90 34 17 34 82 75 98 34 27 69 31 75 93 45 48 3713 59 84 76 59 13 47 69 43 17 91 34 75 93 41 75 90 74 1734 15 74 91 35 79 57 42 39 57 49 02 35 74 23 57 75 11 355Matrix NotationAUseful SubnotationA2 26Useful SubnotationA8 40Matrix Operationsn Additionn Subtractionn Multiplicationn Inverse7Addition14235867+ =A B+ =Addition14235867+ =612810A B+ =C8Addition ConformabilityTo add two matrices A and B:n # of rows in A = # of rows in Bn # of columns in A = # of columns in BSubtraction14235867- =B A- =9Subtraction14235867- =4444B A- =CSubtraction Conformabilityn To subtract two matrices A and B:n # of rows in A = # of rows in Bn # of columns in A = # of columns in B10Multiplication Conformabilityn Regular Multiplicationn To multiply two matrices A and B:n # of columns in A = # of rows in Bn Multiply: A (m x n) by B (n by p)Multiplication General FormulaCij = E Aik x Bkjk=1n11Multiplication I14235867x =A Bx =Multiplication II14235867x =A Bx =C(5x1)C11 = E A11 x B11k=1n12Multiplication III14235867x =A Bx =C(5x1)+(6x3)C11 = E A12 x B21k=2nMultiplication IV14235867x =A Bx =C23 (5x2)+(6x4)C12 = E A1k x Bk2k=1n13Multiplication V14235867x =A Bx =C23(7x1)+(8x3)34C21 = E A2k x Bk1k=1nMultiplication VI14235867x =A Bx =C23 34(7x2)+(8x4)31C22 = E A2k x Bk2k=1n14Multiplication VII14235867x =A Bx =C23 3431 46m x n n x p m x pInner Product of a Vectorn (Column) Vector c (n x 1)c' c2 14x214===c' c21=c214(2x2)+(4x4)+(1x1)15Outer Product of a Vectorn (Column) vector c (n x 1)c c'2 14x214=c c'=c2144 8 28 16 42 4 1Inversen A number can be divided by another number -How do you divide matrices?n Note that a / b = a x 1 / bn And that a x 1 / a = 1n 1 / a is the inverse of a16Unany operations: Inversen Matrix ‘equivalent’ of 1 is the identity matrixn Find A-1such that A-1* A = I1100=IUnary Operations: Inversen Inverse of (2 x 2) matrix¨ Find determinant¨ Swap a11and a22¨ Change signs of a12and a21¨ Divide each element by determinant¨ Check by pre- or post-multiplying by inverse17Inverse of 2 x 2 matrixn Find the determinant= (a11x a22) = (a21x a12)Fordet(A) = (2x3) – (1x5) = 12351=AInverse of 2 x 2 matrixn Swap elements a11and a22Thusbecomes2351=A325118Inverse of 2 x 2 matrixn Change sign of a12and a21Thusbecomes3251=A32-5-1Inverse of 2 x 2 matrixn Divide every element by the determinantThusbecomes(luckily the determinant was 1)32-5-1=A32-5-119Inverse of 2 x 2 matrixn Check results with A-1A =
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