6-Extrema of real valued functions_-1877435609.pdf - 6 Extrema of real valued functions 6.1 Quadratic forms(B24 Textbook Chapter 8 A quadratic form is a

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1 6. Extrema of real valued functions 6.1 Quadratic forms (B24 Textbook Chapter 8) A quadratic form is a degree 2 homogeneous polynomial function. 1 2 , , 1 ( ) ( , , ..., ) n n ij i j i j i j f f x x x u x x = = = x , where not all u ij are zero. f(x) can be written as [ ] 11 12 1 1 22 2 2 1 2 0 ( ) , , ..., 0 0 n n T n nn n u u u x u u x f A x x x u x       = =       x x x U is called the upper triangular coefficient matrix of the quadratic form. For example, [ ] 1 1 2 3 1 2 3 2 3 1 3 2 ( , , ) , , 1 3 4 4 2 1 x f x x x x x x x x     =       It may be written as [ ] 2 2 2 1 2 3 1 1 2 1 3 2 2 3 3 1 T 1 2 3 2 3 ( , , ) 2 6 3 2 1 2 6 , , 0 3 2 U 0 0 1 f x x x x x x x x x x x x x x x x x x = + + + + +     = =       x x Since x i x j = x j x i , it may be rewritten as