Ordinary Differential Equations 25 | Example for Non-Diagonalizable Matrix

A system of linear differential equations with a non-diagonalizable matrix still has a closed-form solution once the matrix exponential is computed—by converting the matrix to Jordan normal form and exploiting the special structure of each Jordan block. With ẋ = Ax and an initial condition x(0)=x0, the unique solution is x(t)=e^{tA}x0, so the real task is calculating e^{tA} when A cannot be diagonalized.

The approach hinges on Jordan normal form: A is written as a block diagonal matrix whose blocks correspond to eigenvalues, with ones on the superdiagonal inside each block and zeros elsewhere. In the worked example, A is already in Jordan form with two Jordan blocks: one for eigenvalue 2 and one for eigenvalue 3. Even when blocks share the same size, their internal “box” structure differs, and that changes the polynomial factors that appear in e^{tA}.

For each Jordan block, the matrix exponential is computed by splitting the block into a diagonal part plus a nilpotent part. Concretely, for the Jordan block associated with eigenvalue 3 (a 3×3 block in the example), the block is decomposed as D+N, where D is diagonal (all 3’s on the diagonal) and N contains only the ones above the diagonal. The nilpotent matrix N satisfies N^2≠0 but N^3=0, so the exponential’s infinite power series collapses to a finite sum: e^{tN}=I+tN+(t^2/2)N^2. A key technical step is that D and N commute, which allows the factorization e^{t(D+N)}=e^{tD}e^{tN}. Since e^{tD} is just the exponential applied to each diagonal entry, the result becomes a triangular matrix with e^{3t} on the diagonal and additional terms like t e^{3t} and (t^2/2)e^{3t} in the superdiagonal positions.

The same method applies to the Jordan block for eigenvalue 2 (a smaller block in the example). Again, the block splits into a diagonal component and a nilpotent component. Here the nilpotent part squares to zero, so e^{tN} truncates after the linear term: e^{tN}=I+tN. Multiplying by e^{tD} produces another triangular matrix with e^{2t} on the diagonal and a t e^{2t} term in the appropriate superdiagonal entry.

Finally, the full matrix exponential e^{tA} is assembled by combining the exponentials of all Jordan blocks in the correct order. The takeaway is practical: even without diagonalizability, Jordan form turns e^{tA} into manageable algebra—triangular matrices whose entries are products of exponentials e^{λt} and low-degree polynomials in t determined by the size of each Jordan block.