I have been given the matrix $A = \begin{bmatrix} -3 & -2 & 2 \\ 0 & 2 & 0 \\ -4 & -1 & 3 \\\end{bmatrix}$.
I firstly needed to find the matrix $P$ that diagonalises it, so I found the eigenvalues of $A$, the corresponding eigenvectors and then constructed $P$ with columns being those eigenvectors. So, $P=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 2 & 1 \\\end{bmatrix}$ (with the corresponding diagonal matrix $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \\\end{bmatrix}$).
I need to use these results to solve the following system of ODEs:
$$x_1' =-3x_1 -2x_2 +2x_3$$
$$x_2' =2x_2$$
$$x_3' =-4x_1 -x_2 +3x_3$$
where $x_i'=\frac{dx_i}{dt}$.
Now I recognise that the coefficients in the system of ODEs correspond directly to the values of matrix $A$, however I am not sure how to use diagonalisation to solve it? I guess you could set up an equation where y $=A$x where y is a column vector $[x_1', x_2', x_3']$ and x is the column vector $[x_1, x_2, x_3]$, however I'm not sure how to go from here? I'm guessing I would use $P$?
Any guidance would be greatly appreciated.
Best Answer
Using the fact that $A=PDP^{-1}$ we can write the system as x'=$A$x=$PDP^{-1}$x or $(P^{-1}x)^{'}=DP^{-1}x$ so setting $y=P^{-1}x$ we solve $y'=Dy$ (which is easily solved). Then once we know y, use x=$P$y to find x.