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:

[tex]
H=\begin{bmatrix}
a & c -i \ d \\
c + i \ d & b
\end{bmatrix}
[/tex]

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

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

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.

 

[blue_raver22]

I can provide a response to this question. It is possible to write a 2N by 2N matrix in terms of N by N matrices. This can be proven mathematically by using the Kronecker product, which is a way of constructing a larger matrix from smaller matrices. The Kronecker product allows us to write a 2N by 2N matrix as a block matrix, with each block being an N by N matrix.

To understand this concept, let's start with two N by N matrices, A and B. The Kronecker product of A and B is denoted as A⊗B and is defined as follows:

A⊗B = [a11B a12B ... a1NB
a21B a22B ... a2NB
... ... ...
aN1B aN2B ... aNNB]

In other words, the Kronecker product of A and B is a block matrix where each block is a scalar multiple of B, with the scalar being the corresponding element of A. This can be extended to larger matrices, such as a 2N by 2N matrix.

To write a 2N by 2N matrix in terms of N by N matrices, we can use the following formula:

M = A⊗I + I⊗B

where M is the 2N by 2N matrix, I is the N by N identity matrix, and A and B are N by N matrices. This can also be extended to more than two matrices, such as:

M = A⊗I⊗C + I⊗B⊗D

where M is a 2N by 2N by 2N by 2N matrix, I is the N by N identity matrix, and A, B, C, and D are N by N matrices.

In terms of a numerical proof, we can use MATLAB or any other programming language to construct a 2N by 2N matrix using the Kronecker product. This would involve creating the smaller N by N matrices, performing the Kronecker product, and then arranging the resulting blocks into a larger matrix.

For further reading on the Kronecker product and its applications in matrix algebra, you can refer to the following resources:

1. "Matrix Computations" by Gene H. Golub