I sit for quite a while now for the following exercise. Let D and E be diagonal matrices, T and S transformation matrices which are invertable. A is a matrix which can be transformed to $A=TDT^{-1}$
I have to show that if $A=TDT^{-1}=SES^{-1}$ it follows that $Tf(D)T^{-1}=Sf(E)S^{-1}$, where f is a function in the complex field that takes in a matrix instead of a variable. It is defined as: $f(A)=Tf(D)T^{-1}$, so that f(D) is a matrix with diagonal entries equal to $f(d_{11})$, for example. So far I deduced that if the first equation holds, we get XD=EX, where X denotes the matrix one gets by $S^{-1}T$. I then showed that is X is invertible and rewrote that as $D=XEX^{-1}$. I tried to insert that into $f(D)=f(XEX^{-1}$) which leads to $Xf(E)X^{-1}$. From this step I do not know how to continue. Thank you for any hints!
$E=S^{-1}TDT^{-1}S$, with $\left(S^{-1}T\right)^{-1}=T^{-1}S$, so by definition $f(E)=S^{-1}Tf(D)T^{-1}S$, and the result follows.