How to Solve System of Equations with Matrix Y

[karush]

nmh{383}
307.27.3 Solve the system
$$Y'=\begin{bmatrix}
1 & 3 & -3 \\
0 & 1 & 0 \\
6 & 3 & -8
\end{bmatrix}Y$$
ok how do we get Y

 

[jvicens]

Hello,

To solve this system, we will use the method of diagonalization. First, we need to find the eigenvalues and eigenvectors of the matrix A = [[1,3,-3],[0,1,0],[6,3,-8]].

The characteristic equation of A is det(A-λI) = 0, where I is the identity matrix. Solving this equation, we get the eigenvalues λ = 1, 1, -2.

Next, we need to find the corresponding eigenvectors. For λ = 1, we have the eigenvector [1,0,1]. For λ = -2, we have the eigenvector [0,1,2].

Now, we can construct the diagonal matrix D = [[1,0,0],[0,1,-2],[0,0,-2]] and the matrix of eigenvectors P = [[1,0,0],[0,1,2],[1,0,1]].

Using the formula Y = P*D*P^-1, we can find the solution to the system Y(t) = e^(At)Y(0).

I hope this helps. Let me know if you have any further questions.