Practice 4 Solutions

3 2 a 2x2 4xy y2 xy 2 2 21 2 2 21 3 5 x y 4

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Unformatted text preview: othogonal linear forms. λ = −3, 1 − x2 − 4xy − y2 = A= xy −1 −2 −2 −1 =− −1 −2 −2 −1 x y 3 2 11 11 =− + 1 2 1 −1 −1 1 3 1 ( x + y) 2 + ( x − y) 2 2 2 [4] Express the quadratic form 2x2 − 4xy + 5y2 as a sum of squares of othogonal linear forms. λ = 1, 6 2x2 − 4xy + 5y2 = A= xy 2 −2 −2 5 2 −2 −2 5 = 1 5 42 21 x y = + 6 5 1 −2 −2 4 1 6 (2x + y)2 + (x − 2y)2 5 5 , 2 × 2 Exercise Set I (quadratic forms), November [5] Express the quadratic form 2x2 + 4xy − y2 as a sum of squares of othogonal linear forms. λ = −2, 3 2 2 2 −1 A= 2x2 + 4xy − y2 = 2 2 2 −1 xy 2 5 =− x y 1 −2 −2 4 =− + 3 5 42 21 3 2 (x − 2y)2 + (2x + y)2 5 5 [6] Express the quadratic form 3x2 + 2xy + 3y2 as a sum of squares of othogonal linear forms. λ = 2, 4 31 13 A= 3x2 + 2xy + 3y2 = 1 −1 −1 1 = 31 13 xy x y 11 11 +2 = ( x − y) 2 + 2 ( x + y ) 2 [7] Express the quadratic form − 2x2 + 4xy + y2 as a sum of squares of othogonal linear forms. λ = −3, 2 A= − 2x2 + 4xy + y2 = xy −2 2 21 =− −2 2 21 3 5 x y 4 −2 −2 1 =− + 2 5 12 24 2 3 ( 2x − y ) 2 + ( x + 2y ) 2 5 5 [8] Express the quadratic form − x2 − 8xy − y2 as a sum of squares of othogonal linear forms. λ = −5, 3 − x2 − 8xy − y2 = A= xy −1 −4 −4 −1 =− −1 −4 −4 −1 x y 5 2 11 11 =− + 3 2 1 −1 −1 1 5 3 ( x + y) 2 + ( x − y) 2 2 2 , 2 × 2 Exercise Set J (recurrence relations), November 2 × 2 Exercise Set J (recurrence relat...
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