Matrix operation to exponentiate each element in a vector

Solution 1:

The "vector exponential" of ${\bf x} := \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}^\top$ can be obtained as follows

$$ \exp\left( \mbox{diag} ({\bf x}) \right) \,{\bf 1}_n$$

Solution 2:

$\bf{X}\boldsymbol{\beta}$ is a vector with 4 rows an 1 column. We denote this vector $\bf{X}\boldsymbol{\beta} = \begin{bmatrix} (\bf{X}\boldsymbol{\beta})_1 \\ (\bf{X}\boldsymbol{\beta})_2 \\ (\bf{X}\boldsymbol{\beta})_3 \\ (\bf{X}\boldsymbol{\beta})_4 \\ \end{bmatrix}$ So,

$\boldsymbol{\beta}^{\textsf{T}}\mathbf{X}^{\textsf{T}} = \left(\bf{X}\boldsymbol{\beta}\right)^\textsf{T} = \begin{bmatrix} (\bf{X}\boldsymbol{\beta})_1 \\ (\bf{X}\boldsymbol{\beta})_2 \\ (\bf{X}\boldsymbol{\beta})_3 \\ (\bf{X}\boldsymbol{\beta})_4 \\ \end{bmatrix}^\textsf{T} $ $ = \begin{bmatrix} (\bf{X}\boldsymbol{\beta})_1 & (\bf{X}\boldsymbol{\beta})_2 & (\bf{X}\boldsymbol{\beta})_3 & (\bf{X}\boldsymbol{\beta})_4 \end{bmatrix}$.

If you want the row vector obtained by making each of the four values in this last matrix the exponent of the number $e$, you can write this as

$$\begin{bmatrix} \exp(\bf{X}\boldsymbol{\beta})_1 & \exp(\bf{X}\boldsymbol{\beta})_2 & \exp(\bf{X}\boldsymbol{\beta})_3 & \exp(\bf{X}\boldsymbol{\beta})_4 \end{bmatrix} =\sum_{k=1}^4 \left[\exp(\bf{e_k}^\intercal\cdot\bf{X}\boldsymbol{\beta})\right] \bf{e_k^{\;\intercal}} $$ where $\bf{e_k}$ is the $k^{th}$ column of the identity matrix (or equivalently, $\bf{e_k^{\;\intercal}}$ is the $k^{th}$ row of the identity matrix).