Groups, Matrices, and Vector Spaces pp 85-111 | Cite as

# Matrix Inverses, Matrix Groups and the \({ LPDU}\) Decomposition

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## 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.

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© Springer Science+Business Media LLC 2017