Spectrum of some polynomial of matrix

Question

Suppose A is a matrix, f(x) is some polynomial. I proved that $f(spectrum(A)) \subseteq spectrum(f(A)$. Can we always say, that f(spectrum A) = spectrum f(A)?

Answer 1

To be more precise than the comments, over an algebraic closed field $k$ (like $\mathbb{C}$), for any matrix $A \in M_{n}(k)$ and any polynomial $f$, you can write $A$ like $PTP^{-1}$ where $P$ is an invertible matrix and $T$ is triangular with diagonal coefficients $\lambda_1,\cdots,\lambda_n$. Then $f(A)=P f(T) P^{-1}$ and $f(T)$ is triangular with diagonal coefficients $f(\lambda_1),\cdots,f(\lambda_n)$. So here $$spec(f(A))=\{ f(\lambda_1),\cdots,f(\lambda_n) \} = f( \{ \lambda_1,\cdots,\lambda_n \} )=f\left( spec(A) \right).$$

But for instance over $\mathbb{R}$, the matrix $A=\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)$ has no real eigenvalue, so $spec_{\mathbb{R}}(A)= \emptyset$ but $A^2+I=0$, so $spec_{\mathbb{R}}(f(A))= \{ 0 \}$ where $f(x)=x^2+1$.