Let A be an n times n symmetric matrix and let f Rn rightarr

Let A be an n times n symmetric matrix and let f: R^n rightarrow R be defined by f(x) = x^tAx. a. Find the gradient f(x). b. Find the Hessian^2f(x). c. Under what conditions on A is the Hessian^2f(x) non-negative definite?

Solution

Note: As the equation editor is not working, we use words to describe Gradient and Hessian

Considet the case n=2. The general case is similar.

a)Let A

=

Then f(v) = A^\' vA = ax² + 2bxy+cy² , where v is the vector (x,y)

So gradient of f = (f_x ,f_y) = (2ax+aby, 2by+cy²)       

                                      = 2Av (the matrix A operating on the vector v)

(b) Hessian of f is the matrix

= 2A

(c) From (b) , Hessian is non-negative definite iff A is so

a	b
b	c