# What does this physics paper mean by having a matrix in a denominator?

I am not familiar with any notation used in physics. The paper "Non-hermitian random matrix theory: Method of hermitian reduction" by Joshua Feinberg and A. Zee (Nuclear Physics B, 1997) states:

A basic tool in studying hermitian random matrices $$\phi$$ (henceforth all matrices will be taken to be $$N\times N$$ with $$N$$ tending to infinity unless otherwise specified) is the Green function defined by $$G(z) = \left< \frac{1}{N} \text{tr} \frac{1}{z - \phi}\right>,$$ where $$\langle \dots \rangle$$ denotes averaging over the random distribution from which the matrices $$\phi$$ are drawn. Diagonalizing $$\phi$$ by a unitary transformation we have $$G(z) = \left< \frac{1}{N} \sum_{k=1}^N\frac{1}{z - \lambda_k}\right>,$$ where the $$N$$ real numbers $$\{ \lambda_k \}$$ are the eigenvalues of $$\phi$$.

In the second equation, everything makes sense. But in the first equation, there is the strange notation of a matrix in the denominator of a fraction, and subtracting a matrix from a scalar. Working backwards, it seems like the first equation is supposed to mean $$(zI - \phi)^{-1}$$. But why is it not written that way?

• Abuse of notation? These are physicists, after all... Mar 22, 2019 at 20:53
• Hahaha! You made my day. Btw, you’re right. Mar 22, 2019 at 21:38
• I’ve gotta say that I’m a mathematician and would also write “operator-scalar” rather than “operator - scalar.identityoperator...” Mar 22, 2019 at 22:33

Your $$(zI-\phi)^{-1}$$ interpretation is correct. The motivation for avoiding it is for analogy with the scalar/eigenvalue case, where we can freely use $$\frac{N}{D}$$ notation. This is especially important when we think about inverse operators, e.g. a Green's function (aka propagator) such as $$\int\frac{d^4k}{(2\pi)^4}\frac{\exp ik\cdot x}{k^2-m^2+i\epsilon}$$.