Rank complement of diagonalizable matrices using polynomial functions

This report defines the rank complement of a diagonalizable matrix (i.e. a matrix which can be brought to a diagonal form by means of a change of basis) as the interchange of the range and the null space. Given a diagonalizable matrix A there is in general no unique matrix Ac which has a range equal to the null space of A and a null space equal to the range of A, only matrices of full rank have a unique rank complement; the zero matrix. Consequently, the rank complement operation is not a distinct operation, but rather a characterization of any operation which makes an interchange of the range and the null space. One particular rank complement operation is introduced here, which eventually leads to an implementation of rank complement operations in terms of polynomials in A. The main result is that for each possible rank r of A there is a polynomial in A which evaluates to a matrix Ac which is a rank complement of A. The report provides explicit expressions for matrix polynomials which compute a rank complement of a symmetric matrix. These results are then generalized to the case of diagonalizable matrices. Finally, a Matlab function is described that implements a rank complement operation based on the results derived.