Mathematics 524 Homework #3

A function f : R^{n} → R of several variables is linear iff there is a vector b ∈ R^{n} and a number c ∈ R so that

f(x) = b · x + c.

The function q : R^{n}R is quadratic iff there is a symmetric matrix A, a vector b ∈ R^{n} and a number c ∈ R so that

q(x) = c + b · x + \frac{1}{2}Ax · x

Thus in two dimensions in terms of coordinates a linear function looks like

f(x_{1}, x_{2}) = b_{1}x_{1} + b_{2}x_{2} + c

and a quadratic function looks like

q(x_{1}, x_{2}) = c + b_{1}x_{1} + b_{2}x_{2} + \frac{1}{2}(a_{11}x_{1}^{2} + 2a_{12}x_{1}x_{2} + a_{22}x_{2}^{2}).

1. Write the vector x ∈ R^{n} in terms of coordinates: x = (x_{1}, x_{2}, …, x_{n}). Now express the linear and quadratic functions f and q above in terms of the coordinates x_{1}, …, x_{n}.

2. For the linear function f compute the gradient ∇f of f and express your answer both as a vector and in coordinates.

3. For the quadratic function q(x) = c + b · x + \frac{1}{2}Ax · x

(a) Compute the gradient ∇q and express your answer as both a vector and in coordinates.

(b) Compute the Hessian Hq of q and express your answer as a matrix.

(c) Assume the matrix A is nonsingular and find the critical point x^{∗} of q.

(d) What is the Taylor expansion of q about the critical point x^{∗}?