The algebra made of square matrices is typically called the matrix algebra. More precisely, if $F$ is a field (such as the real numbers or the complex numbers) and $n$ is a positive integer, then the matrix algebra over $F$ of size $n\times n$ is denoted by $M(n, F)$ or $\mathcal{M}_n(F)$. It consists of all $n\times n$ matrices with entries in $F$, and the operations of addition and scalar multiplication are defined in the usual way.
It is a central algebra.
...
Let's focus on $F=\mathbb C$. The action of the projective linear group $PGL(n, \mathbb C)$ on the matrix algebra is given by conjugation. That is, for any $A$ in the matrix algebra and any $g$ in $PGL(n, \mathbb C)$, the action of $g$ on $A$ is defined as $gAg^{-1}$, where $g^{-1}$ denotes the inverse of $g$. This action is well-defined since scalar matrices are in the center of $GL(n, \mathbb C)$ and are therefore ignored in the quotient.
The automorphisms of this algebra are all inner automorphism (see here, so they are all elements of $PGL(n,\mathbb C)$, because two elements of $GL(n,\mathbb C)$ give rise to the same inner automorphism if they are multiples of each other by a scalar matrix.
Any given 1-parameter group of automorphisms $\{\varphi_t\}$ is, then, a one-parameter subgroup of $PGL(n,\mathbb C)$, and hence is generated by an element of the Lie algebra $\mathfrak{pgl}_n(\mathbb C)$. So there must exist an $A \in \mathcal M_n(\mathbb{C}) \cong \mathfrak{gl}_n(\mathbb{C})$ such that
$$ \varphi_t(B) = \exp(At) B \exp(-At), $$for an arbitrary matrix $B$.
The flow $B(t)=\varphi_t(B)$ satisfies $B(0)=B$ and
$$ \left. \dfrac{{d}}{{d} t} B(t) \right|_{t=0}=AB-BA=[A,B] $$So the 1-parameter group of automorphisms $\{\varphi_t\}$ gives rise to a derivation $D_{\varphi}$ of the algebra:
$$ B \mapsto D_{\varphi}(B):=\left. \dfrac{{d}}{{d} t} B(t) \right|_{t=0} $$Conversely, all the derivations on this algebra are inner: for every derivation $D$ it exists a matrix $A$ of size $n \times n$, such that $D=[-,A]$.
Then, given a derivation $D$ we can construct a 1-parameter group of automorphisms $\varphi^D_t$ in the following way:
$$ \varphi_t^D(B) = \exp(At) B \exp(-At), $$It turns out that
Compare with flow of observables in CM: the algebra is the algebra of smooth functions, the derivations are vector fields.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: