Maximizing quantity which is a product of matrices/vectors

[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?

 

[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.

 

[Bacle2]

For a more general theory, see, e.g:

http://en.wikipedia.org/wiki/Matrix_calculus

 

[AcidRainLiTE]

Both posts were helpful. Thanks.