## Thursday, March 1, 2012

### Nonsingular and Nondefective Random Matrices

In a previous post, I posed the folllowing question:

Consider again, a random square n by n matrix A, whose entries are restricted to the set of integers {-p, -p+1, ..., 0, ... p-1, p}. Each of the 2p+1 values are equally probable.
• What is the probability that this matrix is nonsingular?
• What is the probability that this matrix is nondefective?
For n = 3 and 4, using a simple Monte Carlo method, this is what I get. As p increases, we approach the continuous distribution of entries asymptotically.
 Blue lines are for n=3, and green for n=4. Triangles and squares denote the probability that a random matrix is nonsingular, and nondefective, respectively
As we can see, both singular and defective matrices are extremely rare for large p. Between the two, defectiveness is a rarer feature.