When using ℒ\mathscr{L}-transform method to solve for transition matrix, necessary to invert matrix (sI−A)(sI-A).

Let p(s)=det⁡(sI−A)=sn+an−1sn−1+...+a1s+a0p(s) = \det(sI-A) = s^n + a_{n-1}s^{n-1} + ... + a_1 s + a_0 be the characteristic polynomial of AA and Γ(s)\Gamma(s) the adjoint matrix of (sI−A)(sI - A). (i.e. Γij(s)\Gamma_{ij}(s) is the cofactor of the jijith element of (sI−A)(sI-A))

Matrix theory gives that inverse is given by (sI−A)−1=Γ(s)p(s)(sI - A)^{-1} = \frac{\Gamma(s)}{p(s)}

Iterative procedure to calculate (sI−A)−1(sI - A)^{-1}.

  1. Start with Kn−1=IK_{n-1} = I (an=1a_n = 1). Then for j=1j=1,
  2. calculate an−j=1jtr⁡(Kn−jA)a_{n-j} = \frac{1}{j} \operatorname{tr}(K_{n-j} A),
  3. calculate Kn−j−1=Kn−jA+an−jIK_{n-j-1} = K_{n-j} A + a_{n-j} I,
  4. increase jj by 11 and return to step 2 if j≤nj \leq n. (Check: K−1=ΘK_{-1} = \Theta)

References

  1. https://crrl.poly.edu/6253/lectures/lect7.pdf
  2. P. E. Sarachik, Principles of Linear Systems, Cambridge Press, 1996, pp. 106-107.
  3. https://people.engr.tamu.edu/spb/courses/linearsystems/ECEN605Lecture07.pdf
  4. https://en.wikipedia.org/wiki/Faddeev–LeVerrier_algorithm