Find real $P$ s.t. $B=P^{-1}AP$

Question

TASK:

$\frac{dx}{dt}=Ax$

Given $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & -3 \\ 1 & 3 &2 \end{bmatrix}$

Find real matrix $P$ s.t change of coordinates $x=Py$ transforms the system to and

$\frac{dy}{dt}=By$

$B=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & -3 \\ 0 & 3 &2 \end{bmatrix}$

Solve explicitly for $y$ hence evaluate solution in terms of $x$

APPROACH:

Firstly $PB=AP$

I tried to solve the 9 simultanouesequations and ended up with matrix where there are 3 free variables of the form:

$P=\begin{bmatrix} -10x & 0 & 0 \\ 3x & y & -z \\ x & z &y \end{bmatrix}$

Then I put this into $P^{-1}AP$ in software "SYMBOLAB" and the asnwer he gave me was $B$. So it turns out $x,y,z$may are free as long as $det(P)\neq0$ And since $det(P)=-10x(y^2+z^2)$, all I know is that $x\neq 0$ and at least one of the $y,z$ is not 0.

Is this the correct answer?

Answer 1

This might work.

\begin{align*} P & =\begin{bmatrix}1 & 0 & 0\\ 0 & a & b\\ 0 & -b & a \end{bmatrix} \end{align*}

Answer 2

Besides the hint given in my comment to a related post, consider that the similarity relation is not unique, because in fact: $$ \mathbf{A = R}\;\mathbf{C}\;\mathbf{R}^{\,\mathbf{ - 1}} \quad \quad \Leftrightarrow \quad \left\{ {\begin{array}{*{20}c} {\mathbf{A} = \mathbf{A}\;\mathbf{A}\;\mathbf{A}^{\,\mathbf{ - 1}} } & \Leftrightarrow & {\mathbf{A = }\left( {\mathbf{A}\;\mathbf{R}} \right)\;\mathbf{C}\;\left( {\mathbf{A}\;\mathbf{R}} \right)^{\,\mathbf{ - 1}} } \\ {\mathbf{C} = \mathbf{C}\;\mathbf{C}\;\mathbf{C}^{\,\mathbf{ - 1}} } & \Leftrightarrow & {\mathbf{A = }\left( {\mathbf{R}\;\mathbf{C}} \right)\;\mathbf{C}\;\left( {\mathbf{R}\;\mathbf{C}} \right)^{\,\mathbf{ - 1}} } \\ \begin{gathered} \mathbf{A} = \lambda \;\mathbf{A}\;\lambda ^{\, - 1} \hfill \\ \mathbf{C} = \mu \;\mathbf{C}\;\mu ^{\, - 1} \hfill \\ \end{gathered} & \Leftrightarrow & {\mathbf{A = }\left( {\lambda \;\mathbf{R}\;\mu } \right)\;\mathbf{C}\;\left( {\lambda \;\mathbf{R}\;\mu } \right)^{\,\mathbf{ - 1}} } \\ \end{array} } \right. $$