Question:
Stability and subspaces of a linear system
Please use a step-by-step method to solve the two following exercises:
Exercise 1:
Consider the linear system x' = Ax, where A is a 2 x 2 matrix with lamda in the diagonal as follows;
A = [lamda, -2]
[1 , lamda],
and lamda is real. Determine if the system has a saddle, node, focus, or center at the origin and determine the stability of each node or focus.
Exercise 2:
Consider the linear system x' = Ax, where A is a 3x3 matrix as follows:
A = [ 1, 0 , 0 ]
[ 0, -1, -1 ]
[ 0 , 1, -1 ]
a) Find the general solution of the linear system.
b) Find the stable, unstable, and the center subspaces E^s, E^u and E^c.