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Engineering Calculus Notes 390

Engineering Calculus Notes 390 - 378 CHAPTER 3 REAL-VALUED...

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Unformatted text preview: 378 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 1. The quadratic form Q(x, y, z ) = 2xy + 2xz + 2yz has matrix representative 011 A = [Q] = 1 0 1 110 with characteristic polynomial −λ 1 1 p(λ) = det 1 −λ 1 1 1 −λ = (−λ) det −λ 1 1 −λ − (1) det = −λ(λ2 − 1) − (−λ − 1) + (1 + λ) 11 1 −λ = (λ + 1){−λ(λ − 1) + 1 + 1} = (λ + 1){−λ2 + λ + 2} = −(λ + 1)2 (λ − 2). So the eigenvalues are λ2 = −1. λ1 = 2, The eigenvectors for λ1 = 2 satisfy v2 +v3 = 2v1 v1 +v3 = 2v2 v1 + v2 = 2v3 from which we conclude that v1 = v2 = v3 : so 1 → − = √ (1, 1, 1). u1 3 For λ2 = −1, we need v1 v2 +v3 = − v1 v1 + v2 + v3 = − v2 = − v3 + (1) det 1 −λ 11 ...
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