How would exponentiation be done with matrices?
For example:
$$ \left[\begin{array}{cc}1&2\\8&7\end{array}\right]^{\left[\begin{array}{cc}7&4\\2&9\end{array}\right]}$$
Where both base and exponent are matrices.
$$ \left[\begin{array}{cc}1&2\\8&7\end{array}\right]^6$$
Where the base is a matrix, and the exponent is not.
$$6^{\left[\begin{array}{cc}7&4\\2&9\end{array}\right]}$$
Where the base is not, and the exponent is a matrix.
Wolfram Alpha doesn't recognize it, which makes me wonder if it is even possible.
Using abel and Peter's comments, I was able to solve my first question (matrix exponentiation with two matrices).
Please let me know if I am incorrect in any of my logic.
$$\text{Using a random example:} \\ a = \begin{bmatrix} 1 & 2\\ 8 & 7 \end{bmatrix} \\ b = \begin{bmatrix} 7 & 4\\ 2 & 9 \end{bmatrix} \\ a^{b} = e^{b\ln(a)} \\ \ln(a) = \begin{bmatrix}\ln(1)&\ln(2)\\\ln(8)&\ln(7)\end{bmatrix} \\ \\ a^b = e^{\begin{bmatrix}7&4\\2&9\end{bmatrix}\times\begin{bmatrix}\ln(1)&\ln(2)\\\ln(8)&\ln(7)\end{bmatrix}} \\ a^b = \begin{bmatrix} 7 & 16\\ 256 & 4782969 \end{bmatrix} $$
Wolfram Alpha corroborates the final exponential arithmetic, $e^{\begin{bmatrix}7&4\\2&9\end{bmatrix}\times\begin{bmatrix}\ln(1)&\ln(2)\\\ln(8)&\ln(7)\end{bmatrix}}$, link here.