# The (i,j) component of a Kronecker product?

1. Jun 17, 2009

### Civilized

By the Kronecker product I mean the ordinary tensor product of matrices. In my case I am only interested in square matrices, in fact I want to compute the nonzero elements of products like XXZZXXZZ where X and Z are 2x2 matrices (in fact they are the pauli matrices e.g. the standard representation of the SU(2) algebra).

If I naively compute a tensor product of ~30 pauli matrices, I excede all the computer memory that is availible to a researcher like me. The product itself is quite sparse, and if I only had to store the non-zero elements into memory then I could work with much larger cases of interest. The solution is a formula that computes the i,j component of the tensor product, it seems so straightforward I could do it myself but I presume it has already been done.

2. Jun 18, 2009

### CompuChip

Well if you are really interested in products of the form XXZZXXZZ then you might be interested to note that the squares XX and ZZ of two Pauli matrices are always diagonal (and in fact, are the unit matrix). Also you might be able to combine this fact with the commutation relations to bring any product of the three Pauli matrices X, Y and Z in the form Xm Yn Zk.

3. Jun 18, 2009

### Civilized

Sorry, I ommited the operand because I thought it was clear from the context, but by XXZZXXZZ I mean $X\otimes X \otimes Z \otimes Z \otimes X \otimes X \otimes Z \otimes Z$, where $\otimes$ denotes the Kronecker product.