Question

Solve the following system by bringing the non-diagonalizable matrix A into a Jordan canonical form. Describe the procedure in detail; Draw approximate phase portrait (for a few initial conditions), both for the auxiliary variable (usually denoted y=[ y 1​

y 2 ] ) and for the original one x=[ x 1​ x 2​​ ] x=Ax; A= [3 -1
1 1 ]

Answer 1

To solve the system, find the eigenvectors, determine if A is diagonalizable, find the Jordan canonical form, and draw the phase portrait.

Step-by-step explanation:

To solve the system by bringing the non-diagonalizable matrix A into a Jordan canonical form, we need to follow these steps:

1. Find the eigenvectors of matrix A by solving the equation A - λI = 0.
2. If the matrix A is not diagonalizable, find the Jordan canonical form by finding the generalized eigenvectors.
3. Write the Jordan canonical form of matrix A and use it to solve the system of linear equations.
4. To draw the phase portrait, we can choose a few initial conditions and plot the trajectories of the auxiliary variable and the original variable.