# Maximizing quantity which is a product of matrices/vectors

1. Apr 17, 2013

### AcidRainLiTE

I am trying to follow the following reasoning:

Given a known matrix A, we want to find w that maximizes the quantity

w'Aw​

(where w' denotes the transpose of w) subject to the constraint w'w = 1.

To do so, use a lagrange multiplier, L:

w'Aw + L(w'w - 1)
and differentiate to obtain

Aw = Lw.​

Thus, we seek the eigenvector of A with the largest eigenvalue. ​

I do not understand how they differentiated w'Aw + L(w'w-1) to get Aw = Lw. Can someone explain to me what is going on at that step?

2. Apr 17, 2013

### fzero

It would probably help to write things down explicitly in terms of components,

$$m = w'Aw + L(w'w - 1) = \sum_{ij} A_{ij} w_i w_j + L \left( \sum_i w_i^2 -1 \right).$$

This is a function of the $n$ variables $w_i$. At an extremum, $\partial m /\partial w_k =0$. If you actually work out this set of equations, you'll see they are the components of the eigenvalue equation that you quoted. You'll need to use the fact that $A$ can be taken to be a symmetric matrix.

3. Apr 17, 2013

### Bacle2

4. Apr 18, 2013