Abstract: This talk combines two ideas: inverse functions and matrix functions.

Given a function f(z), it is a standard procedure to define an inverse function through the solution of the equation f(z)=w. If we denote the inverse function by invf, then we can write z=invf(w). The definition is often complicated by the fact that there are multiple solutions of the equation.

Given a scalar function f(z), there are standard procedures for extending the definition to a matrix argument. For example, integer powers of a matrix, or the exponential of a matrix.

There are two ways in which one can arrive at a definition of the inverse of a matrix function. First, one can define the scalar inverse and then extend the definition to a matrix argument; second, one can extend the definition of the equation f(z)=w to matrices and then consider its solutions.

These definitions are discussed in the context of the Lambert W function, which is the inverse of the function f(z) = z*exp(z).