Suppose, I have a matrix equation $H(A) = K$, where $K$, $A$, and $H(A)$ are non-singular, real matrices. $H(A)$ is some function of $A$, for example $H(A) = A^2 + A^3$. Will I always get a solution for $A$ from this equation, for any possible $H$? If not, a counter example will be really helpful.
Update: As @Sobi commented, now I know that this equation does not hold in general. So my updated question is, under which (minimal) conditions, if there is some, I will have a solution? I think if $H$ is a linear operation, then there probably exists a solution. What can be other conditions instead of this; maybe like $H$ and $K$ need to be positive definite or something.
One sufficient condition for example is that $K$ be diagonalizable with eigenvalues in the range of the analytic function $H: \mathbb{R} \rightarrow \mathbb{R}$. For example with $H(x)=x^2+x^3$, the range is all real numbers, so $H(A)=K$ has a solution for any diagonalizable $K$. If $K=PDP^{-1}$, let $D=(d_1,\ldots, d_n)$ and let $E=(a_1,\ldots,a_n)$ such that $H(a_i)=d_i$ for all $1\leq i \leq n$. Then $A=P EP^{-1}$ is a solution of $H(A)=K$.