# Kronecker sum of more than two matrices?

Tags:
1. Feb 27, 2015

### Earthland

1. The problem statement, all variables and given/known data
The question arises from this quote from wikipedia's article about kronecker product:

Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let Hi be the Hamiltonian of the i-th such system. Then the total Hamiltonian of the ensemble is

I have to write this Htot as a ordinary sum over kronecker products of unity matrix and Hi-s.

2. Relevant equations

Kronecker sum for two matrices is defined as

If A is n × n, B is m × m and Ik denotes the k × k identity matrix.

3. The attempt at a solution

Well, as I undesratnd, now instead of A and B we have simply Hi and there should be sum kind of sum over i. But the Kronecker sum is defined only for a pair of matrices and it isn't commutative, so the order is important. I tried something like this, for three H-s:

But it doesn't look very elegant and I have no idea if this could be true. Any advice?

File size:
3.2 KB
Views:
74
2. Mar 1, 2015

### matteo137

The definition is as follows

$$H_{\text{Tot}} =\left( H_1 \otimes \mathbb{I} \otimes \mathbb{I} \otimes ... \right)+ \left(\mathbb{I} \otimes H_2 \otimes \mathbb{I} \otimes ... \right) + \left(\mathbb{I} \otimes \mathbb{I} \otimes H_3 \otimes ... \right) + ...$$

3. Mar 1, 2015

### Earthland

Are you sure? Thank you!

4. Mar 1, 2015

### matteo137

Proof:
Kronecker sum is associative.

In other words.
The Kronecker sum of two matrices is, as you wrote,
$$X=A\oplus B = A\otimes\mathbb{I}_B + \mathbb{I}_A\otimes{B}$$

Now, since the sum $A\oplus B$ is a matrix, $X$, the Kronecker sum
$$Y= X\oplus C = X\otimes\mathbb{I}_C + \mathbb{I}_X\otimes C = (A\otimes\mathbb{I}_B + \mathbb{I}_A\otimes{B})\otimes\mathbb{I}_C + \mathbb{I}_X\otimes{C}$$
Of course $\mathbb{I}_X=\mathbb{I}_A\otimes\mathbb{I}_B$, which gives
$$Y= A\otimes\mathbb{I}_B\otimes\mathbb{I}_C + \mathbb{I}_A\otimes B\otimes\mathbb{I}_C + \mathbb{I}_A\otimes\mathbb{I}_B\otimes{C}$$

$Z= Y\oplus D =$ Keep going... :)

5. Mar 1, 2015

### Earthland

Thank you very much!