Engineering Calculus Notes 381

Engineering Calculus Notes 381 - + is unchanged. Thus, any...

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3.9. THE PRINCIPAL AXIS THEOREM 369 example, there are four kinds of quadratic terms in the two variables x and y : x 2 , y 2 , xy and yx (for the moment, let us ignore the fact that we can combine the last two). Then in the expression Q ( x,y ) = αx 2 + βxy + γyx + δy 2 we can factor out the initial x factor from the Frst two terms and the initial y factor from the last two to write Q ( x,y ) = x ( αx + βy ) + y ( γx + δy ) which can be written as the product of a row with a column = b x y B ± αx + βy γx + δy ² . The column on the right can be expressed in turn as the product of a 2 × 2 matrix with a column, leading to the three-factor product = b x y B ± α β γ δ ²± x y ² . The two outside factors are clearly the coordinate column of ( x,y ) and its transpose. The 2 × 2 matrix in the middle could be regarded as a matrix representing Q , but note that there is an ambiguity here: the two “mixed product” terms βxy and γyx can be rewritten in many other ways without changing their total value; all we need to do is to make sure that the sum
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Unformatted text preview: + is unchanged. Thus, any other matrix with the same diagonal entries and , and whose o-diagonal entries add up to + , leads to the same function Q ( x,y ). To standardize things, we require that the matrix be symmetric. This amounts to balancing the two mixed product terms: each is equal to half will have some useful consequences down the road. Thus the matrix representative of a quadratic form Q ( x,y ) in two variables is the symmetric 2 2 matrix [ Q ] satisfying Q ( x ) = [ x ] T [ Q ][ x ] . (3.30) You should conFrm that this is the same as the matrix representative we used in 3.8 . When we apply Equation ( 3.30 ) to a quadratic form in three variables Q ( x 1 ,x 2 ,x 3 ), we get a symmetric 3 3 matrix. The diagonal entries of [ Q ]...
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