Inversion of a matrix function

Question

Given the function $f:\mathbb{R}^{m \times m} \rightarrow \mathbb{R}^{m \times m}$, over symmetric matrices, defined by $f(A) = A^2 + A^3$.

Do you have any idea of for which values of the range $Y=f(A)$, the function is invertible?

Thanks in advance

Answer 1

Since symmetric matrices are diagonalizable, i.e. for each such $A$ there is a diagonal matrix $D$ and an invertible matrix $S$ such that $$ A = S^{-1}DS, $$ it follows that $$ f(A) = S^{-1} f(D) S. $$ Given $Y$ in the range of $f$, we determine its diagonalization $Y=S^{-1}\tilde DS$. If $f^{-1}(\tilde D)$ is a singleton, $f^{-1}(\tilde D)=\{D\}$, then $X:=S^{-1}DS$ is the unique solution $f(X)=Y$.

Since $f(D)$ only acts on the diagonal elements, the question boils down to find the real numbers $x$ for which $f^{-1}(x)$ is a singleton. It happens that this is the case for all $x<0$ or $x>4/27$, which are local minimum and maximum of $f$.

Hence the function $f$ is invertible for all symmetric matrices, whose eigenvalues belong to the set $(-\infty,0) \cup (4/27,+\infty)$.