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What fraction of square matrices are singular?

  1. Mar 22, 2012 #1
    I was wondering what fraction of 3*3 square matrices are singular? I guess if the elements are real then the answer will be vanishingly small.
    If however, the elements are integers, is there a way to work this number out?
  2. jcsd
  3. Mar 23, 2012 #2


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    I don't think there is any way to get a fraction or percentage. There are countably infinitely many 3x3 matrices with integer entries. And countably many of them are singular. Your "fraction" would be ##\frac \infty \infty##.
  4. Mar 23, 2012 #3
    I don't know the answer to the original question, but this statement seems a poor argument that such a fraction cannot be well defined. There is a well defined sense in which half of all integers are even, despite the sets of both integers and even integers being countably infinite.

    Admittedly the limits might be a little trickier to handle in this case.
  5. Mar 23, 2012 #4
    @kurros: Yeah, if the question is posed as
    'What is the probablility that a given 3*3 matrix with integer entries is singular?" then the answer is either between 0 and 1 or tending to zero.
    It is zero if for every singular matrix, one can find an infinite number of unique non-singular matrices. But I cannot think of a way to prove that.
  6. Mar 23, 2012 #5


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    3x3 matrices over a field form a 9 dimensional vector space over that field and singular ones form an 8 dimensional non linear subvariety of degree 3, defined by setting the determinant equal to zero. a lower dimensional subvariety has density zero if the field of definition is algebraically closed. But if you work over a finite field then you can compute the cardinality of both sets and find a fraction. But over the complex numbers, the probability of a randomly chosen 3x3 matrix being singular is zero.
  7. Mar 23, 2012 #6


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    Hey Bavid.

    One suggestion I have for you if you want to pursue this is to first restrict the range to a finite number of cases.

    For simplicity let N be a fixed number. Then consider all the 3x3 matrices with 0 < X < N where any element may take on the value of X in your matrix.

    From this you can start to get empirical data which might give a pattern for some function of N. From this you could proceed to find a relationship of some sort for N.

    I would also suggest that you make use of number theory since for a singular matrix, at least one row has to be a linear combination of the other and I imagine some parts of number theory could help you analyze these kinds of situations since we are talking about linear relationships over say any non-linear relationship.
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