TITLE: Largest eigenvalue of a $A^T A$ matrix?
QUESTION [5 upvotes]: I have a large real matrix A of size $40K\times 400K$, is there an efficient way to calculate the largest eigenvalue of $A^T A$ (size $400K\times 400K$)?
Thanks.

REPLY [9 votes]: Assuming you want the eigenvalue of largest magnitude (not the largest positive eigenvalue) the most efficient algorithm is power iteration: pick an initial vector $v_0$, then iterate
$$v_i = A^T\left(A\frac{v_{i-1}}{\|v_{i-1}\|}\right).$$
Then $\|v_i\|$ will converge almost surely to the largest magnitude eigenvalue of $A^TA$.
EDIT: Notice that computing $A^TA$ is exceedingly expensive and does not need to be done; the utility of power iteration is that it finds the largest eigenvalue using only matrix-vector products. I've added parentheses to clarify.
EDIT 2: Of course, since $A^TA$ is symmetric positive-semidefinite, the eigenvalues are nonnegative and the largest magnitude eigenvalue is also the largest eigenvalue.