I know that generalized binomial theorem for matrices, i.e. expansion of $$(A+B)^r\quad\quad (*)$$ when $A$ and $B$ are positive-definite matrices and $r$ is a real number, works only when $A$ and $B$ commute. For example, since the identity matrix commutes with all matrices, we have the expansion in this post: Does Newton's generalized binomial theorem work on a matrix?.
My question is that when $A$ and $B$ do not commute, is this possible to expand (*) in the special case that $r$ is very close to 1, i.e. $r=1+\delta$ and $\delta$ is a real number which is very close to zero?
Well, purely formally, at least in physics, one routinely blunders into: $$ (A+B)^{1+\delta}=(A+B) e^{\delta \ln (A+B)}=(A+B) \left (I+\delta \ln (A+B) + \frac{\delta^2}{2} (\ln (A+B))^2+...\right ) , $$ until the alarm bells start ringing.
But, beyond this, you might need to know something about the norm of ln(A+B) to go any further, or else that of (A+B-I), so as to expand the log, formally, $$ \ln (A+B)= \ln (I +(A+B-I))= A+B-I-\frac{1}{2}(A+B-I)^2+... $$ using $\ln (1+x)= x-x^2/2+...$ for "small" x, cf. WP, or the eigenvalues of the sum, etc... Smallness of the matrix A+B-I corresponds to a small norm, as utilized in physics and engineering.