Engineering Calculus Notes 388

Engineering Calculus Notes 388 - 376 CHAPTER 3. REAL-VALUED...

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376 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Now, using both properties given by Proposition 3.9.2 we can use the coordinates with respect to the eigenvectors of [ Q ] to obtain a particularly simple and informative expression for the quadratic form Q : Corollary 3.9.3. If B = { −→ u 1 , −→ u 2 , −→ u 3 } is the basis of unit eigenvectors for the matrix representative A = [ Q ] of a quadratic form, with respective eigenvalues λ i , i = 1 , 2 , 3 , then the value Q ( −→ x ) of Q at any vector can be expressed in terms of its coordinates with respect to B as Q ( −→ x ) = λ 1 ξ 2 1 + λ 2 ξ 2 2 + λ 3 ξ 2 3 (3.32) where B [ −→ x ] = ξ 1 ξ 2 ξ 3 or in other words ξ i = −→ u i · −→ x . The expression for Q ( −→ x ) given by Equation ( 3.32 ) is called the weighted squares expression for Q . We note in passing that the analogous statements hold for two instead of
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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