Can a 2N by 2N matrix written in terms of N by N matrices?

[sokrates]

I posted this question over at the QM page,

https://www.physicsforums.com/showthread.php?t=714076

but I realized I am really looking for a

hard Mathematical proof ...

A description of a numerical way of proving this would also be very helpful for me.

or a reference covering the subject.

Many thanks in advance,

 

[fzero]

As a warmup, would you know how to prove that $\{ I_{2}, \sigma_i\}$ is a basis for Hermitian $2\times 2$ matrices? The result for $2N\times 2N$ will follow by writing the matrix in block form and using the basis as explained by wle in that thread.

 

[sokrates]

Yes - I can do the 2x2 proof I guess.

Because any 2x2 Hermitian matrix can be written as:

$$H=\begin{bmatrix} a & c -i \ d \\ c + i \ d & b \end{bmatrix}$$

where a,b,c,d are all real numbers.

Then H can be uniquely defined in terms of Pauli matrices:
$$\frac{1}{2}\left[ (a+b) \ I_{2\times 2} + (a-b) \ \sigma_z + 2 \ c \ \sigma_x + 2 \ d \ \sigma_y\right]$$

But how to extend this to 2N by 2N ?

 

[sokrates]

Yes, I got it ...

Just write it out explicitly and choose A,B,C,D accordingly to get the random 2N by 2N matrix.

Many thanks for directing me to that.