Hi,

I apologize if this is obvious or has been treated before (I have only found the outer product of vectors being discussed in this aspect). I want to compute the trace of a matrix-matrix product, i.e. given two (large) square matrices A,B, I want to compute tr(AB). Would Eigen be able to use expression templates on this and only evaluate the row-vector products necessary for the diagonal elements of the product? What is the right way to implement this?

Addition to the first question: B is in fact very sparse, only few columns contain non-zero elements. Can this be exploited automagically, i.e. only computing the very few row-column products that would evaluate to non-zero diagonal entries in the product; and, further, would the trace function recognize the diagonal of the matrix is sparse and sum only over the non-zero entries?

Thanks!
-Marc

Hi,

with a not too old version of Eigen, (A*B).diagonal().sum() with A and B dense will be fully optimized (no useless computation, no temporary).

If one is sparse, then (A*B).diagonal() will be assumed to be dense and taking the sum will boils down to something like:

for(int k=0; k<diag_size;++k)
trace += A.row(k).dot(B.col(k));

This dot product will iterate on the non-zeros only, but we cannot turn this dense for loop into a sparse one because the SparseMatrix representation is designed for non-empty columns.