By reading this post you’ll be able to comprehend the basic mechanism behind the proof of LU decomposition Schur Inversion Lemma The Sherman-Morrison-Woodbury inversion formula Small perturbations can’t reduce rank (p216-217 of Meyer) Rank of a matrix equals the rank of its largest nonsingular submatrix (p214 of Meyer; see also Exercise 4.5.15) Characteristic polynomial of $AB$ and $BA$ is the same for square $A,B$ (see Exercise 7.1.19, 6.2.16 and eq. (6.2.1) of Meyer) And many more theorems and lemmas (a few…