Question: Consider the quadratic form f(x) = x + x3 + x3 + 4x1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A

Consider the quadratic form f(x) = x} + x2 + xz + 4x1x2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that f(x) =

Consider the quadratic form f(x) = x + x3 + x3 + 4x1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that f(x) = x' Ax. (b) Find an orthogonal matrix Q so that the change of variables x = Qy transforms the quadratic for f(x) into one with no cross-product terms, that is, diagonalize the quadratic form f(x). Give the transformed quadratic form. (c) Find a vector x of length 1 at which f(x) is maximized. (d) Is the form f(x) positive definite? Explain your answer. (e) Describe the set of points (x1, X2, X3) E R satisfying the equation x + x3 + x3 + 4x1x2 + 4x2x3 + 4x3X1 = 0. That is, name the shape in standard position and find the vector about which it is centrally symmetric. You may also find the vector about which it has been rotated as in the slides.

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