Addition to a random matrix element

[ekkilop]

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!

 

[mfb]

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.

Okay.

and that no row or column has just a single non-zero element -1.

That follows from the sum condition.

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 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.