Question

4. Problem 4. Consider the following system of first order coupled ordinary differential equations, where r (t) and a) Rewrite the initial value problem (IVP) in a matrix form aAi, where ? r (0) +v()() b) Find the three distinct (real) eįgrivalus {A] c) Verify that, satisfies the IVP where the constant ακ fficients c1 c2 and C3 can be detennined from the three given initial conditions. P BIVPn initial 5. Problem 5 (challenge problem): Sinultaneous diagonalization of commuting matrices in the presence of degenerate eigenvalues Find a simultancous basis of eigenvectors for the two (3 x 3)-commuting matrices A and B given by, -(210 )-B-G-1 0) 0 0 () 20),and B=1 1 1 0 , A respectively. In particular, a) Verify that AB = BA; b) Coinpute a diagonalizing matrix MA for A, and verify that AD = M,' AMA; c) Compute a diagonalizing matrix Mp for B, and verify that Bo = Mal BMI: d) Verify that MA'BMA and Mp AMn are block-diagonal (there is a 2 x 2 block and a 1 x 1 block) but not diagonal matrices (therefore, neither MA nor Mp are siumltancous diagonalizing matrices for both A and AD. Hint: Write MM, where At is the diagonalizes the 2 x 2 block in M AMp and leaves the nstruct AM such that M-BM- Bp and M-1AM (3 x 3)-matrix constructed in such a manner that remaining 1 x 1 block in Mp AMp unchanged. f) Please, write down in an explicit manner a simultancous basis of eigenvectors for the two (3 x 3)-commuting matrices A and B

Answer 1

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