# Thread: Vector Maximization

1. ## Vector Maximization

I have a host of questions that require me to maximise vectors/matrices. This is from an economic module so much of this material is new to me. Anyway there is a typical question below. Any examples that I can find in textbooks/online consider the problems of a positive definite matrix and I am unable to see how this changes the nature of the problem.

Consider the problem of maximising $\displaystyle x^T Ax$ subject to $\displaystyle x^Tx=1$ where x is a column vector of n variables and A is a negative-definite, symmetric n-by-n constant matrix. Show that the maximum value of the objective function is equal to the largest eigenvalue of A.

Any help would be much appreciated

2. Originally Posted by AmberLamps
I have a number of questions that require me to maximise vectors/matrices. This is from an economic module so much of this material is new to me. Anyway there is a typical question below. Any examples that I can find in textbooks/online consider the problems of a positive definite matrix and I am unable to see how this changes the nature of the problem.

Consider the problem of maximising $\displaystyle x^T Ax$ subject to $\displaystyle x^Tx=1$ where x is a column vector of n variables and A is a negative-definite, symmetric n-by-n constant matrix. Show that the maximum value of the objective function is equal to the largest eigenvalue of A.

Any help would be much appreciated
i don't think we need $\displaystyle A$ to be negative-definite. we only need $\displaystyle A$ to be (real) symmetric. then there exist an orthonormal basis of $\displaystyle \mathbb{R}^n,$ say $\displaystyle \{x_1, \cdots , x_n \},$ which consists of eigenvectors of $\displaystyle A.$

so we have $\displaystyle Ax_i=\lambda_i x_i,$ for some $\displaystyle \lambda_i \in \mathbb{R}.$ let $\displaystyle \lambda_k= \max_i \lambda_i.$ now if $\displaystyle x \in \mathbb{R}^n$ is any vector with $\displaystyle x^Tx=1,$ then $\displaystyle x=\sum_{i=1}^n \alpha_i x_i,$ for some $\displaystyle \alpha_i \in \mathbb{R}$ and $\displaystyle \sum_{i=1}^n \alpha_i^2=x^Tx=1.$ but then:

$\displaystyle x^TAx=\sum_{i=1}^n \alpha_i^2 \lambda_i \leq \lambda_k \sum_{i=1}^n \alpha_i^2=\lambda_k.$ it's also clear that $\displaystyle x_k^TAx_k=\lambda_k x_k^T x_k=\lambda_k. \ \Box$

3. Thanks, that helps a lot. Don't suppose you know of any good textbooks for this topic?

4. Originally Posted by AmberLamps
Thanks, that helps a lot. Don't suppose you know of any good textbooks for this topic?
if this is a part of a course that you're taking, you should ask your instructor to give you some references. anyway, i guess you can find these stuff in non-elementary linear algebra textbooks.