The Unapologetic Mathematician

Mathematics for the interested outsider

Elementary Matrices Generate the General Linear Group

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September 4, 2009 - Posted by | Algebra, Linear Algebra


  1. […] that if we restrict to upper shears we can generate all upper-unipotent matrices. On the other hand if we use all shears and scalings we can generate any invertible matrix we want (since swaps can be built from shears and scalings). […]

    Pingback by Shears Generate the Special Linear Group « The Unapologetic Mathematician | September 9, 2009 | Reply

  2. HI!
    Very helpful reading this. But how do you prove that you only need the first and third kind of elementary matrices to generate the general linear group?

    Comment by Nicolas | October 5, 2010 | Reply

    • I think I’ll leave that as an exercise: write an arbitrary matrix of the second kind in terms of matrices of the first and third kinds,

      Comment by John Armstrong | October 5, 2010 | Reply

  3. […] all invertible matrices can be written as a product of elementary matrices of the first and third kind, we just have to show that we can row-reduced into the identity using shears before scales, […]

    Pingback by Scales and Shears are (Sort of) Commutative and the Latter Generate the Special Linear Group « indefiniteintegirl | October 26, 2011 | Reply

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