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Identity element for matrices

  1. Sep 19, 2009 #1
    For a set S, there is an identity element e with respect to operation * such that for an element a in S: a*e = e*a = a.

    For a matrix B that is m x n, the identity element for matrix multiplication e = I should satisfy IB = BI = B. But for IB, I is m x m, whereas for BI, I is n x n. Doesn't this mean that the two identity elements are different?
     
  2. jcsd
  3. Sep 19, 2009 #2

    HallsofIvy

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    If m is not equal to n, then, for b an m x n matrix, b*a maps a m dimensional vector to an n dimensional vector. a*b maps an n dimensional vector to an m dimensional vector. for m not equal to n, b*a= a and a*b= a are both impossible and there is no identity matrix.
     
  4. Sep 19, 2009 #3
    So for such cases, would we just say there is a right identity matrix and a left identity matrix? If this was the case, then wouldn't it also imply that the identity mapping is not unique (x2 - x2 + x = x - x + x)? I was thinking (just right now) that the identity mapping simply depended on the form I(x) = x, in which a mapping is defined through the correspondence between a domain and codomain rather than the process through which the mapping occurs. In the same sense, would we say that an identity matrix is simply a matrix that follows the Kronecker delta form regardless of its dimensions?
     
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