Decoupling a system of linear equations

[jjark24]

Hi guys, so I have this system of linear equations:
x' = -3x - 1y
y' = 3x - 7y

And I'm supposed to decouple them by this linear change of variables:
z = Ax + By
w = Cx + Dy

I'm supposed to find values for A, B, C, and D and I have no idea where to begin. Can anyone walk me through this? Thanks!

 

[tiny-tim]

welcome to pf!

hi jjark24! welcome to pf!

essentially, you're looking for the eigenvectors of the matrix, with z' = Pz and w' = Qw

one way of finding them is by writing out z' = Pz in full (in terms of x and y), and then equating coefficients of x and of y (separately)