The Differential in Onto and Inverse Matrix

Unlike injectivity, onto functions do not directly correlate with the differentiability of the function and vice versa, but onto condition is crucial for any function to be invertible. Thus the differentiability of inverse matrix functions depends heavily on the surjectivity of the function. Additionally, a function being invertible and differentiable can lead us to conclude effectively that the inverse of the function is also differentiable. In essence, the graph of the inverse matrix function is a smooth continuous one with unique tangents at every point. 

Surjectivity in Functions

Consider a function F mapped from X to Y. Then we can understand surjectivity as the condition where every element in the co-domain of the function F, that is Y, has a preimage in the domain of F that is X. 

This is an underlying condition for a function to be well-defined and have an inverse ultimately. We can say that the inverse of a function exists only when the function is one-to-one (injective) and onto (surjective). If one of the conditions is not met, we cannot have a well-defined inverse function as a many-one function will give a one-many relation inverted, which is technically not a function, to begin with. Similarly, if the actual function is not onto, then the inverse would not be a complete function as all the elements in the domain of the inverse function would not have an image in the co-domain of the inverse function.

Derivative of an Inverse Function

We just understood the conditions for a function to have an inverse. We now try to find if these inverse functions are differentiable or not. Consider a function F(X) such that it is invertible and differentiable. By extension, we can say that it is continuous and has an inverse. In this scenario, the inverse is probably differentiable.

If F-1(X) is differentiable at X=a, then it must be the case that its differential is the reciprocal of F’(F-1(a)). If we denote the inverse of F(X) and G(X), then G’(X)=1/F’(G(X)). This formula can be directly used to find the differential of the inverse of any function.

Derivative Matrix

Until this point, we saw that the derivative of a function, say F(X)=Y, is the rate of change of the dependent variable Y as the independent variable X changes. This can be a simple 1X1 matrix where the derivative of F(X) at X=a can be written as: DF(a) = [df/dx at a].

However, as we move to multivariate cases, where there are multiple independent variables and multiple functions, we have to form an mXn matrix wherein the rows would be partial derivatives of each component Fi(X) and the columns are scalar-valued functions.

The ordinary differential here is defined as the summation of partial derivatives of the function concerning all independent variables.

The Inverse of a Matrix

Building upon the multivariate situation, consider that there exists a non-singular mXm matrix A whose elements are partial derivatives of a scalar function concerning m number of independent variables.

We know that the product of a matrix and the inverse of the matrix give us an identity matrix I, A-1A=I. If we differentiate both sides of this equation concerning a scalar parameter alpha, then the partial derivative of the inverse of a matrix is equal to the negative value of the inverse matrix multiplied by the derivative of matrix A concerning alpha multiplied by the inverse of the matrix again.

It is necessary to remember here that the inverse of a matrix exists only when its determinant is a non-zero value. Taking a two-variable (x1 and x2) and a two-function (f1 and f2) case and forming a partial derivative matrix, its inverse would only exist when ∂f1/∂x1 * ∂f2/∂x2 – ∂f1/∂x2 * ∂f2/∂x1 is a non zero value. In other words, the product of the partial derivatives of f1 and f2 for x1 and x2 respectively should not be equal to the product of the partial derivatives of f1 and f2 concerning x2 and x1, respectively, for the inverse of the matrix to exist.

Conclusion

We can infer from this discussion that the differentiation of multivariate functions can be highly simplified using the system of vectors and matrices. The identity property of an inverse of a matrix leads us to a direct formula for calculating the differential of the inverse of the matrix. Herein we can refer to the partial derivative matrix as the derivative of the function F at point a. Furthermore, this derivative of the function F can be seen as a linear transformation associated with the matrix DF(a).