How is this operation (Operator to matrix) called?

Question

How do you call this operation?

Create a matrix from operator, with a given orthogonal function system.

I heard this is used in Quantum Mechanics.

Usually this works on linear operators and creates an infinite matrix.

Answer 1

An operator on a vector space $V$ over a field $\mathbb{K}$, if linear, can be given a matrix representation with entries in $\mathbb{K}$

$$A=\varphi\circ\textbf{A}\circ\varphi^{-1}\tag{1}$$

where $\textbf{A}$ is the operator, $A$ its matrix representation under the linear coordinate system $\varphi$ on $V$.

Now the set of all operators on $V$ is an algebra over $\mathbb{K}$ with the composition as algebra product, as is the set of all matrices given by $(1)$ with the matrix product as algebra product. For a fixed $\varphi$, $(1)$ is an algebra isomorphism and so, since $\textbf{A}$ operates on $V$, $A$ acts on $V$ through coordinatization (on the algebra of operators induced by the coordinatization $\varphi$ on $V$).

So $A$ is given by $\textbf{A}$ under $\psi$ (where $\psi\textbf{A}=\varphi\circ\textbf{A}\circ\varphi^{-1}$) that is, under a coordinate system: this is the name of the operation you were looking for.