Reference no: EM131220769

1. Let x ∈ R². Consider the function f(x) = xTAx. If A is an invertible matrix, then prove that this function has only one stationary point at 0(stationary points are the points at which the gradient is zero). Give an A for which 0 is the minimizer of f(x). Give an A for which0 is neither a maximizer nor a minimizer of f(x).

2. Let x ∈ R^p and find the gradient of the following functions:(a) f1(x) = (xT Ax)2, where A is an n × n matrix. (b) f2(x) = (xT Ax)n, where A is an n × nmatrix.

3. Let A ∈ R^n×p denote a fat matrix, i.e., n < p. Explain why we should expect the equation y= Ax to have infinitely many solutions. Among all those solutions we would like to find the one with minimum Euclidean norm, i.e., we want to find the solution with the smallest xT x. Find that solution and prove your answer.