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# Exponential matrix

Let . We define

It is known that this series converges. Prove that

Solution

We triangulise the matrix , that is where is an invertible matrix and is an upper triangular. This is possible since our matrix is over and thus its characteristic polynomial splits. Let be its eigenvalues. Then we note that is upper triangular with in its diagonal. Hence is also upper triangular with in its diagonal. Hence

However and forall . Thus and finally