Pseudo inverse - out of memory

I have got the following equation:

Where chat and λhat are vectors and (B^T*B), H, B^T*Y and 0 are block matrices. The '+' sign indicates the Moore-Penrose pseudo inverse.

The matrix of which the pseudo inverse has to taken has size 4000 x 4000 and is a sparse matrix.

I am able to use an iterative solver to find chat. However, I would like to obtain the diagonal elements of matrix C1 as well. Unfortunately I cannot use Matlab's pinv function because the matrix is too large. I also tried changing svd into svds inside the pinv function so I am able to work with sparse matrices, but then I still get the out of memory error.

Any ideas?