# What is the matrix element?

Can someone give me an Eli5 description of what the matrix element is, particularly in regards to Fermi's Golden Rule?

Fermi's golden rule describes the likelihood of a transition per unit time. Does the matrix element describe the the likelihood of a transition itself without the time element?

Is the matrix element just a variable/operator that facilitates the description of the finite spatial extent of a scattering body?

Or does the matrix element just describe the number of final states available for a given interaction/transition?

The matrix element wiki wasn't very helpful.

Consider for example, a specific case in solid state physics where an electric field is the perturbation. This perturbation allows for transitions of electrons from one a state in one band (i) to a state in another band (f).In the dipole approximation, the material field interaction is given by the product of the electric field amplitude and the dipole moment between the final and initial states. Here the dipole moment $<f|\vec{r} |i>$ is simply the matrix element and its value tells us whether the transition is allowed and its strength.