Matrix Inverses, Matrix Groups and the ${ LPDU}$ Decomposition

Abstract

In this chapter we continue our introduction to matrix theory, beginning with the notion of a matrix inverse and the definition of a matrix group. For now, the main example of a matrix group is the group $GL(n,\mathbb {F})$ of invertible $n\times n$ matrices over a field $\mathbb {F}$ and its subgroups. We will also show that every matrix $A\in \mathbb {F}^{n\times n}$ can be factored as a product ${ LPDU}$, where each of L, P, D, and U is a matrix in an explicit subset of $\mathbb {F}^{n\times n}$. For example, P is a partial permutation matrix, D is diagonal, and L and U are lower and upper triangular respectively.