**When we have functions of matricies, eg. \exp(kA), where k is a scalar and A is a matrix, how is that envisioned? Does the function operate on each element of the matrix individually?**

No, the function does not act on each element of the matrix separately. You have to expand the function in a power series so that you can write the function of a matrix as a sum of terms that involve powers of that matrix. A matrix M to the n-th power is the consecutive matrix multiplication of n copies of the matrix M. See Boas Eqn. 6.17 for the example of \exp(kM) where k is a scalar.

\exp \left ( \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \right ) \neq \left [ \begin{array}{cc} \exp(a) & \exp(b) \\ \exp(c) & \exp(d) \end{array} \right ]

\left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ]^3 = \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \neq \left [ \begin{array}{cc} a^3 & b^3 \\ c^3 & d^3 \end{array} \right ]

\exp(kA) = 1 + kA + \frac{k^2A^2}{2} + \frac{k^3A^3}{3!} + \cdot , where it is understood that A^3 is the matrix multiplication of A A A.