I Generalized Eigenvalues of Pauli Matrices

[thatboi]

Consider a generic Hermitian 2x2 matrix $H = aI+b\sigma_{x}+c\sigma_{y}+d\sigma_{z}$ where $a,b,c,d$ are real numbers, $I$ is the identity matrix and $\sigma_{i}$ are the 2x2 Pauli Matrices. We know that the eigenvalues for $H$ is $d\pm\sqrt{a^2+b^2+c^2}$ but now suppose I have the matrix $\tilde{H} = (\sigma_{x} \otimes A) + (\sigma_{y} \otimes B) + (\sigma_{z} \otimes C)$ , where $\otimes$ denotes the Kronecker product and $A,B,C$ are now N x N diagonal matrices with diagonal entries $a_{i},b_{i},c_{i}$ respectively. I'm wondering if there is some nice generalization of the 2x2 eigenvalue formula in my first statement? I feel like there must be.

 

[Haborix]

I did some half-baked thinking and scratch work and think it is tractable. Seems like the answer is $c_i \pm \sqrt{a_i^2+b_i^2}$. The matrix can be written as something that looks like:

$\tilde{H}=diag(a_1\sigma_x+b_1\sigma_y+c_1\sigma_z, \ldots,a_N\sigma_x+b_N\sigma_y+c_N\sigma_z)$

So it is just a bunch of copies of the simple case.