The exponential of a matrix

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I have an exponential that is raised to the power of a 2x1 matrix and I am wondering what to do with it.

The exact reference I have refers to $e^{\theta u}$ where $u = {\begin{bmatrix}a\\z\end{bmatrix}}$

Any ideas?

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If this is the paper you are talking about, then the matrix exponentials in that paper (like in eq. (3)) are actually scalar exponentials. The column matrix $u_t$ is left-multiplied by a row matrix $\theta_t^T$, and the result $\theta_t^T u_t$ is a number, not a matrix.