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Wasn't sure if I should post this to Linear Algebra or here.

My question is really simple:

Can a 2N by 2N random, and Hermitian Matrix ( Hamiltonian ) be always written as:

[itex]H = A \otimes I_{2\times 2} + B \otimes \sigma_x + C \otimes \sigma_y + D \otimes \sigma_z[/itex]

where A,B,C,D are all N by N matrices, while the sigma's are the Pauli spin matrices.

My question is, as long as A,B,C,D are random and complex Hermitian matrices of size N by N, do I cover the

whole 2N by 2N complex Hermitian space with this representation?

If yes, do you know a reference, a theorem, or a simple proof of this?

A very simple case is when N = 1 , and I know that any 2 x 2 complex , Hermitian matrix can be written as a linear combination of Pauli Matrices.

Many thanks,

sokrates.