exponentiating a matrix and sum of elements

Question

$$ M= \begin{bmatrix} 1&1&0\\0&1&1\\0&0&1\\ \end{bmatrix} $$ Then the sum of all entries of $e^{M}$ i just don't know how to calculate this sum as this would be an infinite series.please help

Answer 1

Hint We can write $$M = I_3 + J_3,$$ where $I_3$ is the identity matrix and $$J_3 := \pmatrix{0&1&0\\0&0&1\\0&0&0}$$ ($J_3$ is the Jordan block of eigenvalue zero and size $3$). Since the summands commute (of course, $I_3$ commutes with every matrix), we have $$\exp M = \exp(I_3 + J_3) = \exp I_3 \exp J_3.$$ Both factors can be computed readily, as (1) $I_3$ is diagonal and (2) $J_3$ is nilpotent, so that all of the terms the infinite series $$\exp J_3 = \sum_{k = 0}^{\infty} \frac{1}{k!} J^k$$ after some point are zero.

Note that $M$ is the Jordan block of eigenvalue $1$ and size $3$; one can generalize the above approach to compute the matrix exponential of any Jordan block and then, since $$\exp(A \oplus B) = \exp A \oplus \exp B \qquad \textrm{and} \qquad \exp(P^{-1} A P) = P^{-1} (\exp A) P,$$ all (squares) matrices.