I have a real, positive-definite, symmetric matrix $M$, calculated from a matrix $A$ which is known:
$M = A^\top A$
Given the nature of $M$, is there any specialised (read: fast code and/or simple code) way to find the largest eigenvalue of $M$? It's years since I did linear algebra, so I'm a bit rusty. I've had a look in Numerical Recipes, Wikipedia and the Matrix cookbook.
I found the following in the Matrix cookbook and got excited too early:
For a symmetric, positive-definite matrix $A$:
$eig\left(A A^\top\right) = eig\left(A^\top A\right) = eig\left(A\right) \cdot eig\left(A\right)$
Of course, it is $M$ that is positive-definite and symmetric, not $A$. $A$ isn't even square. M will probably be 1024x1024 or larger. The size of $M$ (i.e. width of $A$) can be constrained by the algorithm if needed, i.e. I don't mind if your suggestions require that it be a multiple of 4 or a power of 2.
[Edit:] I'm mainly looking for something that I can code natively (taking advantage of SSE/AVX where possible), although your MATLAB/Octave suggestions will definitely be useful to aid my understanding of the algorithms!