Date of Award
Any square matrix A can be decomposed into a sum of the diagonal (DA) and nilpotent (NA) parts as A = DA + NA. The components DA and NA commute with each other and with A. For many matrices A; B, if B commutes with A, then B is a polynomial in A; this holds for DA and NA. Following a Herbert A. Medina preprint, this paper shows how to construct the polynomials p(A) = NA and q(A) = DA. Further, the Jordan canonical form J is a conjugate QAQ^-1 of A; this paper demonstrates that the conjugation relating J and A also relates NA and NJ and DA and DJ, respectively.
Wilson, Kathryn, "A Polynomial in A of the Diagonalizable and Nilpotent Parts of A" (2017). Honors Senior Theses/Projects. 145.