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.
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Notes
- 1.
Named after Carl Friedrich Gauß (1777–1855). A similar method was already described in Chap. 8, “Rectangular Arrays”, of the “Nine Chapters on the Mathematical Art”. This text developed in ancient China over several decades BC stated problems of every day life and gave practical mathematical solution methods. A detailed commentary and analysis was written by Liu Hui (approx. 220–280 AD) around 260 AD.
- 2.
Charles Hermite (1822–1901) .
- 3.
David Hilbert (1862–1943) .
- 4.
The concept of the rank was introduced (in the context of bilinear forms) first in 1879 by Ferdinand Georg Frobenius (1849–1917).
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Liesen, J., Mehrmann, V. (2015). The Echelon Form and the Rank of Matrices. In: Linear Algebra. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-24346-7_5
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DOI: https://doi.org/10.1007/978-3-319-24346-7_5
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