Consider the transform $\widehat{b}=T^{1}b$, where $T=\begin{bmatrix}b & Ab & A^2b & \dots & A^{n1}b \end{bmatrix}$ has full rank. Is it possible to find an explicit expression for \begin{equation} \widehat{b}=\begin{bmatrix} \widehat{b}_1 \\\ \vdots \\\ \widehat{b}_n \end{bmatrix} \end{equation} in terms of the coefficients for $det(I\lambdaA)=det(I\lambda\widehat{A})=a_0+a_1\lambda+\dots+a_{n1}\lambda^{n1}+\lambda^n$. Further, is it possible to derive an explicit expresion for $\widehat{A}=T^{1}AT$ in terms of these coefficients?
Let $T^{(i)}=A^{i1}b$ for $i=1,\cdots n$. You can solve $\widehat{b}$ from the equation $T\widehat{b}=b$. By Cramer's rule, $$ \widehat{b_1}=\frac{det[b, T^{(2)}, \cdots, T^{(n)}]} {det T} $$ $$ \widehat{b_2}=\frac{det[b, b, T^{(3)}, \cdots, T^{(n)}]} {det T} $$ $\cdots$ $$ \widehat{b_n}=\frac{det[b, T^{(2)}, \cdots, T^{(n1)},b]} {det T} $$ Then you can see that $\widehat{b_1}=1, \widehat{b_i}=0$ for all $i=2,\cdots, n$. Further for $\widehat{A}$, use Cramer's rule to each column vector of $\widehat{A}$, in the equation $T\widehat{A} = AT$. 

