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Erschienen in:
2015 | OriginalPaper | Buchkapitel
5. The Echelon Form and the Rank of Matrices
verfasst von : Jörg Liesen, Volker Mehrmann
Erschienen in: Linear Algebra
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Abstract
In this chapter we develop a systematic method for transforming a matrix A with entries from a field into a special form which is called the echelon form of A. The transformation consists of a sequence of multiplications of A from the left by certain “elementary matrices”. If A is invertible, then its echelon form is the identity matrix, and the inverse \(A^{-1}\) is the product of the inverses of the elementary matrices. For a non-invertible matrix its echelon form is, in some sense, the “closest possible” matrix to the identity matrix. This form motivates the concept of the rank of a matrix, which we introduce in this chapter and will use frequently later on.