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

James B. Carrell ()
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James B. Carrell: University of British Columbia, Department of Mathematics

Chapter Chapter 4 in Groups, Matrices, and Vector Spaces, 2017, pp 85-111 from Springer

Abstract: 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.

Date: 2017
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DOI: 10.1007/978-0-387-79428-0_4

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