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Addition to a random matrix element

  1. May 30, 2013 #1
    Hi all!

    I have no application in mind for the following question but it find it curious to think about:

    Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular. Suppose that no row or column of the matrix is the zero vector and that no row or column has just a single non-zero element -1.

    If we add +1 to a random element in our matrix, is there a way to estimate the probability that this new matrix will be singular? If not generally (and I hardly believe it is possible in the general case), is there any special case of a matrix in which such an estimate is obtainable? It is of course clear that the probability is zero for a 2x2 matrix, but how about for larger matrices?

    Any idea would be interesting to hear. Thank you!
  2. jcsd
  3. May 31, 2013 #2


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    Staff: Mentor


    That follows from the sum condition.

    If you know the matrix: Sure, just check all cases (or use some better algorithm).
    If the matrix is not known, is it distributed randomly in some way?

    Singular matrices are a special case (they have 1 degree of freedom less). If there is no special reason why you expect them, the probability is probably 0.
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