1. Solve for eigenvalues/eigenvectors of A.
solve for det(A-lambda*I)=0, find lambda_1, etc...
Then solve for the augmented matrix (A-lambda_?*I | zero vector) (where ? is the ?th eigenvalue you found.)
(or solve systems of equations manually using the 1st, 2nd,...,last rows of A-lambda_?*I = 0 vector, but this way works easier for larger matrices)
If you get a parameter (or variable) when solving for the systems of eqns to get an eigenvector (ex. you get get infinitely many solutions and you get "t" as a parameter), then you should plug in a value for it to get the final v_?.
Each eigenvector is in the form
x2], where x1 and x2 are the solutions to that system of equations.
2. Your general solution is then:
xvec = c1e^(lambda_1t)v1 + c2e^(lambda_2*t)*v2 + ... + cN*e^(lambda_Nt)e^(lambda_Nlambda_1*t)*v1 + c2*e^(lambda_2*t)*v2 + ... + cN*e^(lambda_N*t)*vN
c1,c2,...cN are the first,second,...last constants corresponding to the first,second,...,last (1st, 2nd,...,Nth) eigenvalue and first,second,...,last (1st, 2nd, ... Nth) eigenvector you found when solving for these
(Note: for constants, leave them as c1/c2/etc in gen. solution to DE. When solving an IVP, solve for these and put xvec=your vector and plug in the value of t into the equation),
and t is the variable in the solution to the DE for xvec.