I am using the following matrix algebra to obtain a vector, however, I eventually need all the resulting elements to be exponentiated.
\begin{equation}
\begin{split}
\boldsymbol{\beta}^{\textsf{T}}\mathbf{X}^{\textsf{T}}&=
\begin{bmatrix}
\beta_1 & \beta_2 & \beta_3 & \beta_4 \\
\end{bmatrix}
\begin{bmatrix}
{x_{11} } & {x_{21} } & \cdots & {x_{41} } \\
{x_{12} } & {x_{22} } & \cdots & {x_{42} } \\
\vdots & \vdots & \ddots & \vdots \\
{x_{14} } & {x_{24} } & \cdots & {x_{44} } \\
\end{bmatrix}\\
&=\begin{bmatrix}
\mathbf{x}_1\boldsymbol{\beta} & \mathbf{x}_2\boldsymbol{\beta} & \mathbf{x}_3\boldsymbol{\beta} & \mathbf{x}_4\boldsymbol{\beta} \\
\end{bmatrix}
\end{split}
\end{equation}
What operation would I use to exponentiate each vector element in $\boldsymbol{\beta}^{\textsf{T}}\mathbf{X}^{\textsf{T}}$ to obtain:
\begin{equation}
\begin{bmatrix}
\exp(\mathbf{x}_1\boldsymbol{\beta}) & \exp(\mathbf{x}_2\boldsymbol{\beta}) & \exp(\mathbf{x}_3\boldsymbol{\beta}) & \exp(\mathbf{x}_4\boldsymbol{\beta}) \\
\end{bmatrix}
\end{equation}
The rationale for asking is because I simply just can't introduce a vector whose elements are exponentiated, so I am thinking along the lines of a Kronecker or Hadamard type operation that can exponentiate all elements of a vector. Hence, what would the notation be?
Best Answer
The "vector exponential" of ${\bf x} := \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}^\top$ can be obtained as follows
$$ \exp\left( \mbox{diag} ({\bf x}) \right) \,{\bf 1}_n$$