Feb 16, 2015

Solution to Student Puzzle Corner 7

The problem, stated here, was the following:
what is an exact expression for the singularity probability $f(p)$ of a $3\times 3$ random matrix with iid Rademacher entries having the distribution $P(1) = p, P(-1) = 1-p$, to find its average over $p$, and to show that it is minimized at $p = \frac{1}{2}$.

Student members Yixin Wang (Columbia University) and Tengyuan Liang (Wharton School, University of Pennsylvania), provided correct solutions to the various parts of the problem.

SPC Yixin Wang Tengyuan Liang
Yixin Wang (left) and Tengyuan Liang both answered correctly

This can actually be done by a brute force complete enumeration. Alternatively, condition on the first two rows and denote the third row by $(x,y,z)$, and calculate the determinant. It is a linear function of $x, y, z$; find the probability that the determinant is zero. Now uncondition and obtain the unconditional probability that the determinant is zero. At first sight, the singularity probability $f(p)$ is a polynomial of degree $9$. But cancellation occurs and it is actually the sixth degree
polynomial

\[ f(p) = 1-18p^2+84p^3-162p^4+144p^5-48p^6.\]

Immediately, the average singularity probability is

$\int_0^1 f(p)\,dp = \frac{26}{35}$, and $f(\frac{1}{2}) = \frac{5}{8}$.

Pictures can be deceiving, and this is an instance. A coarse scale picture would suggest that $f(p)$ is convex over the unit interval. It is not. It has the concave-convex-concave shape.

$f”(p)$ is a polynomial of degree four and has a double zero at $p = \frac{1}{2}$ and two other distinct zeros in the unit interval. It has no other real or complex roots. The first derivative $f'(p)$ is a polynomial of degree five, has a zero at $p = 0, 1,$ and $\frac{1}{2}$, and the zero at $\frac{1}{2}$ is a triple zero. The first derivative has no other roots. From the symmetry of $f(p)$ and the fact that it is strictly decreasing in a right neighborhood of zero and so, strictly increasing in a left neighborhood of $p = 1$, it now follows that the minimum must be at $p = \frac{1}{2}$.

We have here the function $f(p)$ written explicitly; an interesting question is whether the coefficients in the polynomial have some relations to a well known number theoretic function. The same question could be asked for other values of $n$.

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